Wikibooks zhwikibooks https://zh.wikibooks.org/wiki/Wikibooks:%E9%A6%96%E9%A1%B5 MediaWiki 1.45.0-wmf.5 first-letter Media Special Talk User User talk Wikibooks Wikibooks talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk Transwiki Transwiki talk Wikijunior Wikijunior talk Subject Subject talk TimedText TimedText talk Module Module talk 3 24021 181519 139426 2025-06-14T17:55:25Z Aqurs1 50970 Aqurs1移動頁面[[User talk:唐舞麟]]至[[User talk:Astroneering]]：​当重命名用户“[[Special:CentralAuth/唐舞麟|唐舞麟]]”至“[[Special:CentralAuth/Astroneering|Astroneering]]”时自动移动页面 139426 wikitext text/x-wiki <div style="width: 100%; margin: 0.2em 0; padding: 0.3em 0 0.3em 0; border-top:1px solid #DDD; border-bottom:1px solid #DDD; font-size: 120%;">'''您好，{{PAGENAME}}！[[Wikibooks:欢迎|歡迎]]来到[[Wikibooks:关于维基教科书|维基教科书]]！'''</div> [[檔案:Wikibooks-logo.svg|100px|right]] 这里包括[[W:内容开放|内容开放]]的教科书及手册。目前总共有[[Special:Statistics|{{NUMBEROFARTICLES}}]]页教科书及手册。 *在开始进入维基教科书的世界之前，请先浏览'''[[Wikibooks:欢迎|欢迎]]'''以及'''[[Wikibooks:关于|关于]]'''。 *如果您有什么问题欢迎到'''[[Wikibooks:互助客栈|互助客栈]]'''或在我的对话页面提出。亦可以參考'''[[Wikibooks:其他语言版的维基教科书|其他语言版]]'''與'''[[Help:目录|帮助手册]]'''。 *您可以在'''[[Wikibooks:首頁|首頁]]'''按主题浏览或參與編寫您所感興趣的維基教科書。 *编辑後加入某段讨论时请“您的签名和签名时间”，只需打上四个英文波浪号(<nowiki>~~~~</nowiki>)便可。 *最後，我們希望您可以'''[[w:Wikipedia:勇于更新页面|勇于编辑]]'''。希望您会喜欢这里！ <div lang="en" style="font-style: italic; font-size: 90%;">Welcome to Chinese Wikibooks. Thank you for your contributions! If you don't prefer to use Chinese, you may leave a message [[wikibooks:Guestbook|Guestbook]]. When looking for further information, feel free to visit [[Wikibooks:互助客栈|Mutual Aid Center]]. Enjoy!</div> [[Wikibooks:維基教科書人|維基人]]：[[User:逆襲的天邪鬼|逆襲的天邪鬼]] ([[User talk:逆襲的天邪鬼|留言]]) 2017年6月22日 (四) 09:43 (UTC) == 关于侵权的页面 == [[File:Information.svg|25px|link=]] 您好，欢迎来到维基教科书。我们邀请每个人为这个本教科书贡献有建设性的资料，謝謝您願意一起來改進维基教科书。(因为维基教科书方针仍不完善，故以下部分的链接均指向维基百科，二者版权要求相似，可供参考)。维基教科书不可以复制其他網站的文章，可惜的是您所加入的内容'''并不符合[[w:Wikipedia:版权常见问题解答#百度百科的版权问题|百度百科的版权问题的有关规定]]'''。建議您可以利用自己所知，用'''自己的話'''重新組織論述、改寫原先已經有的文章内容，並且將添加或者修改的文字加上參考的出處；這樣應該能夠讓讀者們更容易理解您要表達的意思。希望您下次撰寫教科书時能秉持'''原創'''或'''翻譯其他語言版本的维基教科书'''來作為編寫守則。感谢您对我们的支持。 以上为套话，简单说是我发现[[必修三 政治与法治]] 这个新页面很多内容是复制与百度百科，故挂了侵权模板(PS：请勿随意移除模板)。首先不谈百度百科的可靠性问题，从其他地方复制内容，大多数情况下都有版权问题。以我为数不多的经验，只有同版权协议的文字才能复制过来，并且要加上出处。不懂之处可以加群(Telegram群组为@wikibooks_zh，QQ群为159218217)讨论。总之较为保险的方法是原创、同义转述和翻译（翻译其他语言的教科书）。另：我看您对政治这部分比较感兴趣，如果想原创撰写课本可以参考一下教育部发布的相关课程标准，还是十分有参考价值的。原创毕竟很难，稍微简单一点的方式是翻译。以上是一些建议供您参考，祝编辑愉快。——[[User:Woclass|Woclass]] ([[User talk:Woclass|留言]]) 2018年5月12日 (六) 14:02 (UTC) == 关于对高中数学的编写建议 == 维基教科书的中学教材争论是个老话题了。我觉得应该就写一个比较完善的版本，不需要为不同的地区分开编写。当然，其它的观点我也不会完全反对。 我认为主要有3点理由： # 现在维基教科书基本上没有多少活跃用户，而且都是各自为战的。能坚持把至少一套教程认真写出来就不错了，而后还有多余时间和精力的话，再去争论这个地区差异化的问题也不迟。内容的空缺是目前最大的问题。如果没有足够的内容细节作为支撑，为一个目录怎么安排、取舍而争执，是不可取的。最后可能就是几个人断断续续讨论了大半天，整出来一个或几个版本的目录，结果到最后人力、精力分散，都挤不出足够时间从头到尾地亲自去完善自己理想版本中的那些具体内容，使应该充满干货的教科书沦为空壳烂尾项目。 # 之前的高中数学教科书上按中国大陆某版本的章节划分编排的。先不说内容极度不完善，根本没有人善后。非常难办的是这边教科书开工还没过几年，填坑的时间都遥遥无期，大陆现在又在全面推行新版本的教材，而且网上很多人都觉得新版本比之前的版本内容编排更“合理”，那没写完的烂尾教科书应该怎么办呢？把整个目录推倒重来？（很多重要章节顺序变了，所有例题、习题、讲解肯定都需要逐一检查所涉及的知识点是否符合新的教授顺序，不是直接重命名和移动章节顺序那么简单。）然后又过几年，才写好了少数页面的半成品又根据官方教材和考试政策调整而从头大修？台湾那边更杂，都是用民间出版社的教材，而且取舍、讲授顺序很不统一，有的学校据说有因为嫌弃本地教科书难度低、顺序不够好教授，开始引用大陆教科书。台湾这个情况又怎么处理？ # 中国大陆的中学教育一直都在削减教科书中的知识点，高等教育则不断缩减专业基础课的课时，不知道是怎么想的。大多数只跟着教材走的普通学生肯定会知识面越来越窄，考虑问题的眼界一届不如一届。高中毕业后能力不如老一辈的高中生，大学毕业后能力不如老一辈的大学生。这是开倒车的做法，维基教科书这边当然不应该跟进。使内容跟进考试标准也是没有充足理由的，学习数学的目的不是应付考试，而是思考方法和锻炼解决问题的能力。考试只是检验其中一部分常用的知识点和方法，不是学习的根本目的。我相信许多数学优秀的学生，在中学阶段的所学知识绝不仅仅局限于教科书上的有限的内容。有一部分初等数学的内容中学考试不会考，但是也不是大学数学的内容，因此好的高中教科书应该补充上这样一些初-高中衔接和高中-大学衔接的知识。比如大陆现行的教材都是不会正式介绍三角函数的和差化积公式的，差一点的学校根本不会教授给学生，但是大学微积分学中的三角函数导数公式证明中会用到和差化积公式（有的还不一定会指出其中用到的公式名字），如果学生此前不怎么知道这个公式就只能当作是莫名其妙空降而来的知识了。一些非常有用的公式、方法和二级结论也不会出现在常规考试范围中，这就是考试本身的局限。我认为有意义的人生不能被“考试”这2个字所局限，数学学习也有它作为解题工具之外的其它意义。教育要到了大学阶段才是真正的人才培养，大学以前的教育都只是文化扫盲。所以在中学阶段为部分有能力的学生扩展视野，减少教育资源的不公平分配造成的不公平竞争，从而为大学阶段的学习打好知识基础也是极有意义的。当然，我也不支持数学越难越好，补充什么内容主要按后续大学课程中的实际需要而定。所以编写中学教科书也需要从大学的视角出发，进入大学需要的预备知识不能在中学阶段选择性忽略掉。我的做法就是尽量在每一节的正文之前讲清楚考试范围和要求，内容细节供读者结合自己的需要来自行取舍，但绝不会因为考试不考而回避相关的重要知识点（尤其是对以后很有用的）。大多数人最终是用数学来解决生产、工程、科研、财务中的各种问题，而不是解决各种堆砌技巧与巧合特例的智力难题。数学如果不是为人类文明的前进提供助力、不贴近实际或前沿的需要，而是泛滥于制造精巧的智力难关，那就索然无味了。 最后，数理化的教科书写起来很耗时，文字间的各种公式、符号穿插和十分枯燥、任务量大但是又不得不小心仔细的数字校对，一言难尽。能坚持主干内容原创（习题则不易原创），把至少一个版本写出来就是对人的忍耐力的不小挑战了。我既然有想法付诸行动，必然不希望它烂尾或是几年一次地频繁大修。一本有价值、千锤百炼的免费教科书，应该向着能争取流传几百年的高标准去写，向后人展示承载了我们这个时代教育经验的精选读本，而不是服务于眼前朝令夕改的各地区考试。 -- [[User:Giggle2005|Giggle2005]] ([[User talk:Giggle2005|留言]]) 2020年12月31日 (四) 08:48 (UTC) 印象里一般英美系的教科书比较喜欢讲清楚来龙去脉，各种细枝末节都娓娓道来，优点是照顾新手、容易入门，缺点是篇幅太大，信噪比低。欧陆系（包括罗刹国）的教科书则比较枯燥，但是也简练扼要，思维训练比较深入。就数学学科而言，数学教学比较难趣味化，数学本身就是抽象难度大的学科，就像学习几何学没有捷径，只能一个接一个定理去掌握。当然容不容易贴近生活实际，也要看具体分支。线性代数的话，本身就是应用及广的分支，应用方面的例子容易找，所以也是数学建模的经典工具。有一些分支比如泛函分析（线性或非线性的）、复变函数的话，虽然也是很基础的领域，但是能找到的例子也几乎都是纯数学语言描述的或者有较多其它专业背景的，这类的例子举出来读者也不一定能消化。之所以有人去研究这些看起来枯燥抽象的主题应该主要是出于对数学本身的好奇，应用方面的动机倒像是次要的。有时候为了减少误解（很多结论违法初学者的直观感受），还不如开篇就直接从数学角度描述，前期少牵扯到在其它学科中的具体应用。当然，如何将2种教材风格结合在一起，写出来不知道会是什么样子的，可能会很厚哦。 现在确实我只忙于将各种方法和题目整理出来，而且简单的习题偏少，对入门层次的读者肯定不友好。现在的样子只能当作草稿，定位偏高但是内容干瘪、不接地气，有一些例子也更适合划入习题，导致总体风格有一些四不像，这个我知道。掌握Mathematica和Geogebra这类软件也非常重要，完全可以结合各章节所学的知识在数学软件中做很多有趣的事情，摸索很多不知道的规律。只是限于考试压力，不会有人重视数学信息化的教学，这是很遗憾的。后期有时间的话，我计划把各章加入更多讲解、实例和动画。 说个题外话，美国数学学会的官网就是典型的不重视美观和趣味，很土很单调的设计。这可能和数学家追求大道至简的作风有关。所以数学书如果写得死板，那么不喜欢数学的人会更不喜欢数学，喜欢数学的人倒是无所谓。大陆很多好学校实验班使用的自编校本教材都是没有什么趣味性的，就是典型的只负责把重点题型和知识点讲清楚的内部讲义，没有什么洋洋洒洒、生动有趣的文字解说，但是在启发优等生方面确实效果不错，同时也是学渣们一想到就能吐出来的噩梦。80年代数学家项武义主编的教材，数学理论味就很重，各种在中学考试并不会考的数学思想（比如运算的封闭性、群的思想、线性相关的概念、寻找空间变换下的不变量）贯穿其中，但是也最有特色，能体现数学家对本学科问题的处理方式和数学本来的严肃面目。 中国大陆的现行教材难度确实是过于简化了。虽然尽力照顾了初学者和差生，但是仅仅止步于此。由于内容删减，对往年经典习题又缺乏收录，最后既不能满足中等及以上学生的需要，也根本不能满足实际考试要求。而差生也不一定会认真看书，所以这种教材的实际功效就很尴尬。比如说豆瓣网上“数学必修4”的2则书评： * “课本简单的几页，题却难得要死。” * “这本书编的比较容易，然而考试并不是这么容易啊。” 所以收录一定的中档题还是很有必要的，这就是我主要在做和关心的事情。只是目前收集的简单题还不太多，这个其实在往年教科书中摘录或简单改编一些即可，问题倒是比较好解决。 言归正传。本来我只是打算写别的主题（比如趣味数学或是一些应用数学）的，但是由于一些解题方法和基础知识牵涉到中学数学内容，但是在中文维基百科里面找不到相关内容或某些解题方法不适合写在维基百科里，就只能自己去完善中学数学的内容。等于我是为了知识体系的完整性去填中学数学的坑，所以暂时没有优先考虑易读性、美观性和贴近考试要求，至少近期也不会有兴致写那么完善。如果我打算加入动画，我肯定又会去顺便把Flash、3D Studio Max、Blender这些会用到的软件的教程先搞起来，最后就会精力比较分散。 说到您正在翻译的线性代数，向量的几种基本运算在游戏制作中就都典型应用。叉积可以判断物体在角色的左侧还是右侧、前方还是后方等等，点积可以用于决定角色前方的视野角度，归一化则常用于获取相对方位、防止角色斜着移动时速度更快的问题。如果能将其和Unity 3D或虚幻引擎结合起来，就可以做出很有意思的互动演示。不过我觉得传统教材里将叉积作为矢量处理还是不妥的，它毕竟是一个披着伪矢量外衣的张量，并不满足矢量定义中的指标变换规则，我不知道为什么人们在初学阶段常将其说成是矢量。另外，那个线性代数如果任务量太大，建议不要按从头到尾的顺序翻译，因为很有可能还没熬到精彩章节登场就烦得不想继续了。如果优先翻译精彩的章节，其它章节则先选译一部分必要的段落，这样会比较有意思一些。 相关链接可以先不改动，毕竟我目前不能保证我真的有耐心写完，承认自己有可能中途放弃也不是什么说不出口的事情。如果以后其他人觉得不合适，要搅一棍，后期酌情删改即可。不必牵一发而动全身。 最后，恭喜发财。 -- [[User:Giggle2005|Giggle2005]] ([[User talk:Giggle2005|留言]]) 2020年12月31日 (四) 23:01 (UTC) 6t0u2bl3yuryrcxbwwyub68uo2efmzs 0 25842 181527 181342 2025-06-15T07:37:53Z Д.Ильин 66321 img 181527 wikitext text/x-wiki 我们已在高中初步学习了价层电子对互斥模型，并且了解了价层电子对数的计算公式<math>n= \frac{1}{2}(a-xb)</math>（a为中心原子的族序数，b为配位原子达到稳定结构需要的电子数，x为配位原子的数量）。在这里我们会学习一种新的判断VSEPR模型的方法。 对于以A原子为中心原子的某一分子（或离子）AX_n，我们将其写为AX_nE_m，E表示中心原子A上的孤对电子对数。那么有： <math>m=\frac {1}{2} (a-n|b|-q)</math> 其中a为A的族序数，b为X的化合价，q为离子所带电荷数（带正负号）。 {{例题|计算下列微粒的中心原子上的孤对电子对数：<math>SO_2</math>，<math>SO_3</math>，<math>SO_3^{2-}</math>，<math>SO_4^{2-}</math>。|<math>\begin{array}{|c|c|}particle&m\\\hline SO_2&1\\SO_3&0\\SO_3^{2-}&1\\SO_4^{2-}&0\\\end{array}</math>}}我们令<math>z=m+n</math>，便可写出AY_z这个通式。这里的z表示价层电子对数。这样，我们便可以与高中阶段联系起来了。 z与VSEPR模型对应关系如下表： {| |z |2 |3 |4 |5 |6 |- |VSEPR模型 |直线型 |平面三角形 |正四面体型 |三角双锥型 |正八面体型 |- |球棍模型 |[[File:AX2E0-3D-balls.png|无框|100x100像素]] |[[File:AX3E0-3D-balls.png|无框|100x100像素]] |[[File:AX4E0-3D-balls.png|无框|101x101像素]] |[[File:Trigonal-bipyramidal-3D-balls.png|114x114像素]] |[[File:AX6E0-3D-balls.png|无框|106x106像素]] |} 我们舍去孤对电子，便得到了粒子的实际构型。但孤对电子实际位置需要通过下列几种斥力顺序判断： # 孤对电子对-孤对电子对>孤对电子对-键合电子对>键合电子对-键合电子对 # 双键-单键>单键-单键 # 电负性弱-电负性弱>电负性弱-电负性强>电负性强-电负性强 {{例题|已知四氟化硫<math>SF_4</math>分子，判断其VSEPR模型和空间构型。|2=<math>m=\frac{1}{2}(6-1\times 4)=1</math>，故<math>SF_4</math>属<math>AX_4E_1</math>，z=5。其VSEPR模型为三角双锥形。<br /> 分子中有一对孤对电子，它可能出现的位置有两种——(1)中心原子所在的三角形平面外；(2)中心原子所在的三角形平面内。<br />考虑斥力：(1)分子中有3个90度的孤对电子对-键合电子对，三个90度的键合电子对-键合电子对，三个120度的键合电子对-键合电子对；(2)分子中有两个90度的孤对电子对-键合电子对，两个120度的孤对电子对-键合电子对，四个90度的键合电子对-键合电子对，一个120度的键合电子对-键合电子对。<br /> 由于120度电子对之间的斥力远小于90度电子对，可以忽略不计，所以(2)的斥力比(1)稳定。<br /> 故四氟化硫分子呈变形四面体型（又称跷跷板型），如下图。<br /> [[File:Seesaw-3D-balls.png|120px]]}} 一般来讲，确定了z、m之后，分子构型如下表所示： {| class="wikitable" !电子对数 !m=0 !m=1 !m=2 !m=3 |- |2 |[[File:AX2E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E0-2D.png|128x128像素]] <center>直线型</center> |&nbsp; |&nbsp; |&nbsp; |- |3 |[[File:AX3E0-side-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX3E0-side-2D.png|128x128像素]] <center>平面三角型</center> |[[File:AX2E1-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E1-2D.png|128x128像素]] <center>角型（V型）</center> |&nbsp; |&nbsp; |- |4 |[[File:AX4E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E0-2D.png|131x131像素]] <center>四面体型</center> |[[File:AX3E1-2D.svg|链接=https://zh.wikipedia.org/wiki/File:AX3E1-2D.svg|131x131像素]] <center>三角锥型</center> |[[File:AX2E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E2-2D.png|128x128像素]] <center>角型（V型）</center> |&nbsp; |- |5 |[[File:AX5E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX5E0-2D.png|131x131像素]] <center>三角双锥型</center> |[[File:AX4E1-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E1-2D.png|128x128像素]] <center>变形四面体型（跷跷板型）</center> |[[File:AX3E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX3E2-2D.png|128x128像素]] <center>T型</center> |[[File:AX2E3-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E3-2D.png|128x128像素]] <center>直线型</center> |- |6 |[[File:AX6E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX6E0-2D.png|133x133像素]] <center>八面体型</center> |[[File:AX5E1-2D-1.svg|链接=https://zh.wikipedia.org/wiki/File:AX5E1-2D.png|133x133像素]] <center>四角锥型</center> |[[File:AX4E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E2-2D.png|133x133像素]] <center>平面正方形型</center> |&nbsp;T型 |} pgwgwuoljgiinxm5jm7vrrpfj0055ha 181528 181527 2025-06-15T07:46:26Z Д.Ильин 66321 img 181528 wikitext text/x-wiki 我们已在高中初步学习了价层电子对互斥模型，并且了解了价层电子对数的计算公式<math>n= \frac{1}{2}(a-xb)</math>（a为中心原子的族序数，b为配位原子达到稳定结构需要的电子数，x为配位原子的数量）。在这里我们会学习一种新的判断VSEPR模型的方法。 对于以A原子为中心原子的某一分子（或离子）AX_n，我们将其写为AX_nE_m，E表示中心原子A上的孤对电子对数。那么有： <math>m=\frac {1}{2} (a-n|b|-q)</math> 其中a为A的族序数，b为X的化合价，q为离子所带电荷数（带正负号）。 {{例题|计算下列微粒的中心原子上的孤对电子对数：<math>SO_2</math>，<math>SO_3</math>，<math>SO_3^{2-}</math>，<math>SO_4^{2-}</math>。|<math>\begin{array}{|c|c|}particle&m\\\hline SO_2&1\\SO_3&0\\SO_3^{2-}&1\\SO_4^{2-}&0\\\end{array}</math>}}我们令<math>z=m+n</math>，便可写出AY_z这个通式。这里的z表示价层电子对数。这样，我们便可以与高中阶段联系起来了。 z与VSEPR模型对应关系如下表： {| |z |2 |3 |4 |5 |6 |- |VSEPR模型 |直线型 |平面三角形 |正四面体型 |三角双锥型 |正八面体型 |- |球棍模型 |[[File:AX2E0-3D-balls.png|无框|100x100像素]] |[[File:AX3E0-3D-balls.png|无框|100x100像素]] |[[File:AX4E0-3D-balls.png|无框|101x101像素]] |[[File:Trigonal-bipyramidal-3D-balls.png|114x114像素]] |[[File:AX6E0-3D-balls.png|无框|106x106像素]] |} 我们舍去孤对电子，便得到了粒子的实际构型。但孤对电子实际位置需要通过下列几种斥力顺序判断： # 孤对电子对-孤对电子对>孤对电子对-键合电子对>键合电子对-键合电子对 # 双键-单键>单键-单键 # 电负性弱-电负性弱>电负性弱-电负性强>电负性强-电负性强 {{例题|已知四氟化硫<math>SF_4</math>分子，判断其VSEPR模型和空间构型。|2=<math>m=\frac{1}{2}(6-1\times 4)=1</math>，故<math>SF_4</math>属<math>AX_4E_1</math>，z=5。其VSEPR模型为三角双锥形。<br /> 分子中有一对孤对电子，它可能出现的位置有两种——(1)中心原子所在的三角形平面外；(2)中心原子所在的三角形平面内。<br />考虑斥力：(1)分子中有3个90度的孤对电子对-键合电子对，三个90度的键合电子对-键合电子对，三个120度的键合电子对-键合电子对；(2)分子中有两个90度的孤对电子对-键合电子对，两个120度的孤对电子对-键合电子对，四个90度的键合电子对-键合电子对，一个120度的键合电子对-键合电子对。<br /> 由于120度电子对之间的斥力远小于90度电子对，可以忽略不计，所以(2)的斥力比(1)稳定。<br /> 故四氟化硫分子呈变形四面体型（又称跷跷板型），如下图。<br /> [[File:Seesaw-3D-balls.png|120px]]}} 一般来讲，确定了z、m之后，分子构型如下表所示： {| class="wikitable" !电子对数 !m=0 !m=1 !m=2 !m=3 |- |2 |[[File:AX2E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E0-2D.png|128x128像素]] <center>直线型</center> |&nbsp; |&nbsp; |&nbsp; |- |3 |[[File:AX3E0-side-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX3E0-side-2D.png|128x128像素]] <center>平面三角型</center> |[[File:AX2E1-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E1-2D.png|128x128像素]] <center>角型（V型）</center> |&nbsp; |&nbsp; |- |4 |[[File:AX4E0-2D.svg|链接=https://zh.wikipedia.org/wiki/File:AX4E0-2D.svg|131x131像素]] <center>四面体型</center> |[[File:AX3E1-2D.svg|链接=https://zh.wikipedia.org/wiki/File:AX3E1-2D.svg|131x131像素]] <center>三角锥型</center> |[[File:AX2E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E2-2D.png|128x128像素]] <center>角型（V型）</center> |&nbsp; |- |5 |[[File:AX5E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX5E0-2D.png|131x131像素]] <center>三角双锥型</center> |[[File:AX4E1-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E1-2D.png|128x128像素]] <center>变形四面体型（跷跷板型）</center> |[[File:AX3E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX3E2-2D.png|128x128像素]] <center>T型</center> |[[File:AX2E3-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E3-2D.png|128x128像素]] <center>直线型</center> |- |6 |[[File:AX6E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX6E0-2D.png|133x133像素]] <center>八面体型</center> |[[File:AX5E1-2D-1.svg|链接=https://zh.wikipedia.org/wiki/File:AX5E1-2D.png|133x133像素]] <center>四角锥型</center> |[[File:AX4E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E2-2D.png|133x133像素]] <center>平面正方形型</center> |&nbsp;T型 |} aya5xg9ashewzst889ecfvorrrqhkdy 181529 181528 2025-06-15T08:36:07Z Д.Ильин 66321 img 181529 wikitext text/x-wiki 我们已在高中初步学习了价层电子对互斥模型，并且了解了价层电子对数的计算公式<math>n= \frac{1}{2}(a-xb)</math>（a为中心原子的族序数，b为配位原子达到稳定结构需要的电子数，x为配位原子的数量）。在这里我们会学习一种新的判断VSEPR模型的方法。 对于以A原子为中心原子的某一分子（或离子）AX_n，我们将其写为AX_nE_m，E表示中心原子A上的孤对电子对数。那么有： <math>m=\frac {1}{2} (a-n|b|-q)</math> 其中a为A的族序数，b为X的化合价，q为离子所带电荷数（带正负号）。 {{例题|计算下列微粒的中心原子上的孤对电子对数：<math>SO_2</math>，<math>SO_3</math>，<math>SO_3^{2-}</math>，<math>SO_4^{2-}</math>。|<math>\begin{array}{|c|c|}particle&m\\\hline SO_2&1\\SO_3&0\\SO_3^{2-}&1\\SO_4^{2-}&0\\\end{array}</math>}}我们令<math>z=m+n</math>，便可写出AY_z这个通式。这里的z表示价层电子对数。这样，我们便可以与高中阶段联系起来了。 z与VSEPR模型对应关系如下表： {| |z |2 |3 |4 |5 |6 |- |VSEPR模型 |直线型 |平面三角形 |正四面体型 |三角双锥型 |正八面体型 |- |球棍模型 |[[File:AX2E0-3D-balls.png|无框|100x100像素]] |[[File:AX3E0-3D-balls.png|无框|100x100像素]] |[[File:AX4E0-3D-balls.png|无框|101x101像素]] |[[File:Trigonal-bipyramidal-3D-balls.png|114x114像素]] |[[File:AX6E0-3D-balls.png|无框|106x106像素]] |} 我们舍去孤对电子，便得到了粒子的实际构型。但孤对电子实际位置需要通过下列几种斥力顺序判断： # 孤对电子对-孤对电子对>孤对电子对-键合电子对>键合电子对-键合电子对 # 双键-单键>单键-单键 # 电负性弱-电负性弱>电负性弱-电负性强>电负性强-电负性强 {{例题|已知四氟化硫<math>SF_4</math>分子，判断其VSEPR模型和空间构型。|2=<math>m=\frac{1}{2}(6-1\times 4)=1</math>，故<math>SF_4</math>属<math>AX_4E_1</math>，z=5。其VSEPR模型为三角双锥形。<br /> 分子中有一对孤对电子，它可能出现的位置有两种——(1)中心原子所在的三角形平面外；(2)中心原子所在的三角形平面内。<br />考虑斥力：(1)分子中有3个90度的孤对电子对-键合电子对，三个90度的键合电子对-键合电子对，三个120度的键合电子对-键合电子对；(2)分子中有两个90度的孤对电子对-键合电子对，两个120度的孤对电子对-键合电子对，四个90度的键合电子对-键合电子对，一个120度的键合电子对-键合电子对。<br /> 由于120度电子对之间的斥力远小于90度电子对，可以忽略不计，所以(2)的斥力比(1)稳定。<br /> 故四氟化硫分子呈变形四面体型（又称跷跷板型），如下图。<br /> [[File:Seesaw-3D-balls.png|120px]]}} 一般来讲，确定了z、m之后，分子构型如下表所示： {| class="wikitable" !电子对数 !m=0 !m=1 !m=2 !m=3 |- |2 |[[File:AX2E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E0-2D.png|128x128像素]] <center>直线型</center> |&nbsp; |&nbsp; |&nbsp; |- |3 |[[File:AX3E0-side-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX3E0-side-2D.png|128x128像素]] <center>平面三角型</center> |[[File:AX2E1-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E1-2D.png|128x128像素]] <center>角型（V型）</center> |&nbsp; |&nbsp; |- |4 |[[File:AX4E0-2D.svg|链接=https://zh.wikipedia.org/wiki/File:AX4E0-2D.svg|131x131像素]] <center>四面体型</center> |[[File:AX3E1-2D.svg|链接=https://zh.wikipedia.org/wiki/File:AX3E1-2D.svg|131x131像素]] <center>三角锥型</center> |[[File:AX2E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E2-2D.png|128x128像素]] <center>角型（V型）</center> |&nbsp; |- |5 |[[File:AX5E0-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX5E0-2D.png|131x131像素]] <center>三角双锥型</center> |[[File:AX4E1-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E1-2D.png|128x128像素]] <center>变形四面体型（跷跷板型）</center> |[[File:AX3E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX3E2-2D.png|128x128像素]] <center>T型</center> |[[File:AX2E3-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX2E3-2D.png|128x128像素]] <center>直线型</center> |- |6 |[[File:AX6E0-2D.svg|链接=https://zh.wikipedia.org/wiki/File:AX6E0-2D.png|133x133像素]] <center>八面体型</center> |[[File:AX5E1-2D-1.svg|链接=https://zh.wikipedia.org/wiki/File:AX5E1-2D.png|133x133像素]] <center>四角锥型</center> |[[File:AX4E2-2D.png|链接=https://zh.wikipedia.org/wiki/File:AX4E2-2D.png|133x133像素]] <center>平面正方形型</center> |&nbsp;T型 |} jpx3io0z7s9gz5poptfo5x2o0vpp02z 0 30861 181514 138787 2025-06-14T14:20:06Z Astroneering 39717 重写本节 181514 wikitext text/x-wiki == 生物世界的奥妙：从诗意自然到科学探索 == === 从自然画卷到生命图谱：生物世界的多样性 === 大千世界，纷繁复杂的生命画卷总能引发人类的惊叹与思考。喧鸟覆春洲，杂英满芳甸"的春日盛景，"小荷才露尖尖角，早有蜻蜓立上头"的夏日生机，"一声梧叶一声秋，一点芭蕉一点愁"的秋意萧瑟，"马毛带雪汗气蒸，五花连钱旋作冰"的冬日壮美——这些跨越四季的诗句不仅是文学的凝练，更是对生命多样性的诗意礼赞。从极地冰原到赤道雨林，从深海热泉到高山草甸，生命的足迹遍布地球每个角落。这些景象不仅展现了生物与环境的和谐共生，更揭示了生命科学的核心命题：'''生命的多样性'''（Biodiversity）——从微观的细胞结构到宏观的生态系统，从单细胞的古菌到复杂的哺乳动物，生命以千姿百态的形式编织着地球的活力。 <br> 然而，当我们凝视这幅生命长卷时，疑问也随之而来：为什么北极熊只生活在北极，而企鹅却独属南极？为什么马达加斯加岛上的狐猴与非洲大陆的灵长类如此相似却又截然不同？为什么人类拥有共同的祖先，却演化出多样的族群？这些问题的答案，藏匿于生物演化的史诗中，也镌刻在每一片树叶的叶脉、每一块骨骼的化石、每一串DNA的碱基序列里。 <br> === 生物的起源与演化：从分子到文明的旅程 === 生命的起源犹如宇宙中的星辰碰撞，既充满偶然，又遵循必然规律。现代科学认为，约38亿年前，原始海洋中的无机分子在闪电、火山活动或深海热泉口的高温高压环境下，逐渐形成了氨基酸、核苷酸等有机小分子，最终聚合为原始细胞。这一过程被米勒-尤里实验等实验证实，揭示了生命诞生的化学基础。<br> 从单细胞到多细胞，从水生到陆生，生命的演化如同一部波澜壮阔的史诗。达尔文的《物种起源》提出了自然选择的理论框架，而现代综合进化论则融合了遗传学、分子生物学的成果，解释了突变、基因重组、隔离机制如何驱动物种分化。例如，澳大利亚的袋鼠与欧亚大陆的胎盘哺乳动物的分道扬镳，正是冈瓦纳古陆分裂后生殖隔离的产物；而大熊猫的“伪拇指”——一根膨大的腕骨演化成抓握竹子的工具——则完美诠释了自然选择对现有结构的“妥协性改造”。 <br> === 生物多样性的密码：基因、环境与适应 === 生物形态的差异背后，是基因组的精妙调控。以人类为例，所有个体共享超过99%的DNA序列，但仅0.1%的差异便造就了肤色、体型、代谢特征的多样性。这种变异源于突变、基因重组和环境压力的共同作用。例如，镰刀型细胞贫血症的致病基因在疟疾流行区反而具有生存优势，揭示了“有害基因”在特定环境中的保留机制。 生态位（Ecological Niche）理论进一步解释了生物分布的规律。热带雨林中，每平方公里可容纳上百种树木，而沙漠中的植物则演化出肉质茎叶、夜间开花等特性。这种差异源于能量流动与资源竞争的生态法则：雨林光照充足、水分充沛，为物种共存提供了物理基础；而沙漠的极端环境则筛选出耐旱、节水的生存策略。 === 疾病与健康的博弈：生命系统的脆弱与韧性 === 人类对自身的探索始终充满敬畏。当流感病毒的蛋白质外壳与宿主细胞受体结合时，一场微观的攻防战悄然展开。免疫系统通过T细胞的抗原识别与B细胞的抗体产生构筑防线，而病毒则以高频突变突破封锁。这种“军备竞赛”揭示了生命系统的动态平衡——健康是机体稳态（Homeostasis）与外部压力博弈的结果。 癌症的发生更凸显了生命机制的双刃剑属性。原癌基因的激活与抑癌基因的失活打破了细胞周期调控，使正常细胞转化为恶性增殖的“叛逆者”。这一过程既是个体衰老的必然产物，也是自然选择无法完全清除有害突变的例证。现代医学正试图通过靶向治疗、免疫疗法等手段重建体内平衡，而这离不开对生命本质的深刻理解。 === 生态系统的脆弱性与保护生物学的使命 === 生物分布的差异性背后，是生态系统的层级性规律。以亚马逊雨林为例，其单位面积生物量与物种丰富度远超撒哈拉沙漠，这与两地的能量输入、水循环、土壤肥力等生态因子密切相关。然而，人类活动正以前所未有的速度改变着这一平衡：热带雨林的砍伐导致栖息地破碎化，塑料污染使海洋生物误食致死，气候变化迫使物种向极地迁移。 保护生物学（Conservation Biology）的诞生正是对这一危机的科学回应。通过建立自然保护区、实施濒危物种繁育计划、恢复退化生态系统，人类试图在发展与保护间寻找支点。中国的大熊猫保护工程便是一个典范：通过栖息地连通、人工繁殖放归等措施，这一曾濒临灭绝的物种种群得以恢复。 === 生命科学的未来：从实验室到文明进程 === 21世纪的生命科学正在重塑人类认知。基因编辑技术（如CRISPR-Cas9）让科学家能够精准修改DNA片段，为遗传病治疗带来曙光；合成生物学通过设计人工基因回路，使微生物成为生产药物或生物燃料的微型工厂；而脑科学与人工智能的融合，则试图破解意识的生物学基础。 与此同时，生命科学也面临伦理拷问：我们是否有权编辑人类胚胎基因？人工智能模拟的意识是否具有权利？这些问题的答案，将决定人类如何运用这把“普罗米修斯之火”。 === 结语：走进生命科学的殿堂 === 从细胞膜上的蛋白质泵到全球碳循环，从孟德尔的豌豆实验到人类基因组计划，生命科学始终在追问一个根本命题：**生命为何如此运作**？学习生物学，不仅是为了解生命的奥秘，更是为在技术爆炸的时代做出理性判断，在生态危机的十字路口选择前行方向。翻开这本教材，你将踏上一段穿越时空的旅程——从微观的分子世界到宏观的地球系统，从达尔文的加拉帕戈斯群岛到当代的合成生物学实验室，共同探索生命的本质与人类文明的未来。 2dpp15zop8pv9k8opdecvqihdyy5kzm 181516 181514 2025-06-14T14:29:45Z Astroneering 39717 /* 生物的起源与演化：从分子到文明的旅程 */ 181516 wikitext text/x-wiki == 生物世界的奥妙：从诗意自然到科学探索 == === 从自然画卷到生命图谱：生物世界的多样性 === 大千世界，纷繁复杂的生命画卷总能引发人类的惊叹与思考。喧鸟覆春洲，杂英满芳甸"的春日盛景，"小荷才露尖尖角，早有蜻蜓立上头"的夏日生机，"一声梧叶一声秋，一点芭蕉一点愁"的秋意萧瑟，"马毛带雪汗气蒸，五花连钱旋作冰"的冬日壮美——这些跨越四季的诗句不仅是文学的凝练，更是对生命多样性的诗意礼赞。从极地冰原到赤道雨林，从深海热泉到高山草甸，生命的足迹遍布地球每个角落。这些景象不仅展现了生物与环境的和谐共生，更揭示了生命科学的核心命题：'''生命的多样性'''（Biodiversity）——从微观的细胞结构到宏观的生态系统，从单细胞的古菌到复杂的哺乳动物，生命以千姿百态的形式编织着地球的活力。 <br> 然而，当我们凝视这幅生命长卷时，疑问也随之而来：为什么北极熊只生活在北极，而企鹅却独属南极？为什么马达加斯加岛上的狐猴与非洲大陆的灵长类如此相似却又截然不同？为什么人类拥有共同的祖先，却演化出多样的族群？这些问题的答案，藏匿于生物演化的史诗中，也镌刻在每一片树叶的叶脉、每一块骨骼的化石、每一串DNA的碱基序列里。 <br> === 生物的起源与演化：从分子到文明的旅程 === 生命的起源犹如宇宙中的星辰碰撞，既充满偶然，又遵循必然规律。现代科学认为，约38亿年前，原始海洋中的无机分子在闪电、火山活动或深海热泉口的高温高压环境下，逐渐形成了氨基酸、核苷酸等有机小分子，最终聚合为原始细胞。这一过程被米勒-尤里实验等实验证实，揭示了生命诞生的化学基础。<br> 从单细胞到多细胞，从水生到陆生，生命的演化如同一部波澜壮阔的史诗。达尔文的《物种起源》提出了自然选择的理论框架，而现代综合进化论则融合了遗传学、分子生物学的成果，解释了突变、基因重组、隔离机制如何驱动物种分化。例如，澳大利亚的袋鼠与欧亚大陆的胎盘哺乳动物的分道扬镳，正是冈瓦纳古陆分裂后生殖隔离的产物；而大熊猫的“伪拇指”——一根膨大的腕骨演化成抓握竹子的工具——则完美诠释了自然选择对现有结构的“妥协性改造”。 <br> === 生物多样性的密码：基因、环境与适应 === 生物形态的差异背后，是基因组的精妙调控。以人类为例，所有个体共享超过99%的DNA序列，但仅0.1%的差异便造就了肤色、体型、代谢特征的多样性。这种变异源于突变、基因重组和环境压力的共同作用。例如，镰刀型细胞贫血症的致病基因在疟疾流行区反而具有生存优势，揭示了“有害基因”在特定环境中的保留机制。 生态位（Ecological Niche）理论进一步解释了生物分布的规律。热带雨林中，每平方公里可容纳上百种树木，而沙漠中的植物则演化出肉质茎叶、夜间开花等特性。这种差异源于能量流动与资源竞争的生态法则：雨林光照充足、水分充沛，为物种共存提供了物理基础；而沙漠的极端环境则筛选出耐旱、节水的生存策略。 === 疾病与健康的博弈：生命系统的脆弱与韧性 === 人类对自身的探索始终充满敬畏。当流感病毒的蛋白质外壳与宿主细胞受体结合时，一场微观的攻防战悄然展开。免疫系统通过T细胞的抗原识别与B细胞的抗体产生构筑防线，而病毒则以高频突变突破封锁。这种“军备竞赛”揭示了生命系统的动态平衡——健康是机体稳态（Homeostasis）与外部压力博弈的结果。 癌症的发生更凸显了生命机制的双刃剑属性。原癌基因的激活与抑癌基因的失活打破了细胞周期调控，使正常细胞转化为恶性增殖的“叛逆者”。这一过程既是个体衰老的必然产物，也是自然选择无法完全清除有害突变的例证。现代医学正试图通过靶向治疗、免疫疗法等手段重建体内平衡，而这离不开对生命本质的深刻理解。 === 生态系统的脆弱性与保护生物学的使命 === 生物分布的差异性背后，是生态系统的层级性规律。以亚马逊雨林为例，其单位面积生物量与物种丰富度远超撒哈拉沙漠，这与两地的能量输入、水循环、土壤肥力等生态因子密切相关。然而，人类活动正以前所未有的速度改变着这一平衡：热带雨林的砍伐导致栖息地破碎化，塑料污染使海洋生物误食致死，气候变化迫使物种向极地迁移。 保护生物学（Conservation Biology）的诞生正是对这一危机的科学回应。通过建立自然保护区、实施濒危物种繁育计划、恢复退化生态系统，人类试图在发展与保护间寻找支点。中国的大熊猫保护工程便是一个典范：通过栖息地连通、人工繁殖放归等措施，这一曾濒临灭绝的物种种群得以恢复。 === 生命科学的未来：从实验室到文明进程 === 21世纪的生命科学正在重塑人类认知。基因编辑技术（如CRISPR-Cas9）让科学家能够精准修改DNA片段，为遗传病治疗带来曙光；合成生物学通过设计人工基因回路，使微生物成为生产药物或生物燃料的微型工厂；而脑科学与人工智能的融合，则试图破解意识的生物学基础。 与此同时，生命科学也面临伦理拷问：我们是否有权编辑人类胚胎基因？人工智能模拟的意识是否具有权利？这些问题的答案，将决定人类如何运用这把“普罗米修斯之火”。 === 结语：走进生命科学的殿堂 === 从细胞膜上的蛋白质泵到全球碳循环，从孟德尔的豌豆实验到人类基因组计划，生命科学始终在追问一个根本命题：**生命为何如此运作**？学习生物学，不仅是为了解生命的奥秘，更是为在技术爆炸的时代做出理性判断，在生态危机的十字路口选择前行方向。翻开这本教材，你将踏上一段穿越时空的旅程——从微观的分子世界到宏观的地球系统，从达尔文的加拉帕戈斯群岛到当代的合成生物学实验室，共同探索生命的本质与人类文明的未来。 qfj29rscg4valuhbjbwh68kbntbrm8z 181518 181516 2025-06-14T14:42:54Z Astroneering 39717 /* 结语：走进生命科学的殿堂 */ 181518 wikitext text/x-wiki == 生物世界的奥妙：从诗意自然到科学探索 == === 从自然画卷到生命图谱：生物世界的多样性 === 大千世界，纷繁复杂的生命画卷总能引发人类的惊叹与思考。喧鸟覆春洲，杂英满芳甸"的春日盛景，"小荷才露尖尖角，早有蜻蜓立上头"的夏日生机，"一声梧叶一声秋，一点芭蕉一点愁"的秋意萧瑟，"马毛带雪汗气蒸，五花连钱旋作冰"的冬日壮美——这些跨越四季的诗句不仅是文学的凝练，更是对生命多样性的诗意礼赞。从极地冰原到赤道雨林，从深海热泉到高山草甸，生命的足迹遍布地球每个角落。这些景象不仅展现了生物与环境的和谐共生，更揭示了生命科学的核心命题：'''生命的多样性'''（Biodiversity）——从微观的细胞结构到宏观的生态系统，从单细胞的古菌到复杂的哺乳动物，生命以千姿百态的形式编织着地球的活力。 <br> 然而，当我们凝视这幅生命长卷时，疑问也随之而来：为什么北极熊只生活在北极，而企鹅却独属南极？为什么马达加斯加岛上的狐猴与非洲大陆的灵长类如此相似却又截然不同？为什么人类拥有共同的祖先，却演化出多样的族群？这些问题的答案，藏匿于生物演化的史诗中，也镌刻在每一片树叶的叶脉、每一块骨骼的化石、每一串DNA的碱基序列里。 <br> === 生物的起源与演化：从分子到文明的旅程 === 生命的起源犹如宇宙中的星辰碰撞，既充满偶然，又遵循必然规律。现代科学认为，约38亿年前，原始海洋中的无机分子在闪电、火山活动或深海热泉口的高温高压环境下，逐渐形成了氨基酸、核苷酸等有机小分子，最终聚合为原始细胞。这一过程被米勒-尤里实验等实验证实，揭示了生命诞生的化学基础。<br> 从单细胞到多细胞，从水生到陆生，生命的演化如同一部波澜壮阔的史诗。达尔文的《物种起源》提出了自然选择的理论框架，而现代综合进化论则融合了遗传学、分子生物学的成果，解释了突变、基因重组、隔离机制如何驱动物种分化。例如，澳大利亚的袋鼠与欧亚大陆的胎盘哺乳动物的分道扬镳，正是冈瓦纳古陆分裂后生殖隔离的产物；而大熊猫的“伪拇指”——一根膨大的腕骨演化成抓握竹子的工具——则完美诠释了自然选择对现有结构的“妥协性改造”。 <br> === 生物多样性的密码：基因、环境与适应 === 生物形态的差异背后，是基因组的精妙调控。以人类为例，所有个体共享超过99%的DNA序列，但仅0.1%的差异便造就了肤色、体型、代谢特征的多样性。这种变异源于突变、基因重组和环境压力的共同作用。例如，镰刀型细胞贫血症的致病基因在疟疾流行区反而具有生存优势，揭示了“有害基因”在特定环境中的保留机制。 生态位（Ecological Niche）理论进一步解释了生物分布的规律。热带雨林中，每平方公里可容纳上百种树木，而沙漠中的植物则演化出肉质茎叶、夜间开花等特性。这种差异源于能量流动与资源竞争的生态法则：雨林光照充足、水分充沛，为物种共存提供了物理基础；而沙漠的极端环境则筛选出耐旱、节水的生存策略。 === 疾病与健康的博弈：生命系统的脆弱与韧性 === 人类对自身的探索始终充满敬畏。当流感病毒的蛋白质外壳与宿主细胞受体结合时，一场微观的攻防战悄然展开。免疫系统通过T细胞的抗原识别与B细胞的抗体产生构筑防线，而病毒则以高频突变突破封锁。这种“军备竞赛”揭示了生命系统的动态平衡——健康是机体稳态（Homeostasis）与外部压力博弈的结果。 癌症的发生更凸显了生命机制的双刃剑属性。原癌基因的激活与抑癌基因的失活打破了细胞周期调控，使正常细胞转化为恶性增殖的“叛逆者”。这一过程既是个体衰老的必然产物，也是自然选择无法完全清除有害突变的例证。现代医学正试图通过靶向治疗、免疫疗法等手段重建体内平衡，而这离不开对生命本质的深刻理解。 === 生态系统的脆弱性与保护生物学的使命 === 生物分布的差异性背后，是生态系统的层级性规律。以亚马逊雨林为例，其单位面积生物量与物种丰富度远超撒哈拉沙漠，这与两地的能量输入、水循环、土壤肥力等生态因子密切相关。然而，人类活动正以前所未有的速度改变着这一平衡：热带雨林的砍伐导致栖息地破碎化，塑料污染使海洋生物误食致死，气候变化迫使物种向极地迁移。 保护生物学（Conservation Biology）的诞生正是对这一危机的科学回应。通过建立自然保护区、实施濒危物种繁育计划、恢复退化生态系统，人类试图在发展与保护间寻找支点。中国的大熊猫保护工程便是一个典范：通过栖息地连通、人工繁殖放归等措施，这一曾濒临灭绝的物种种群得以恢复。 === 生命科学的未来：从实验室到文明进程 === 21世纪的生命科学正在重塑人类认知。基因编辑技术（如CRISPR-Cas9）让科学家能够精准修改DNA片段，为遗传病治疗带来曙光；合成生物学通过设计人工基因回路，使微生物成为生产药物或生物燃料的微型工厂；而脑科学与人工智能的融合，则试图破解意识的生物学基础。 与此同时，生命科学也面临伦理拷问：我们是否有权编辑人类胚胎基因？人工智能模拟的意识是否具有权利？这些问题的答案，将决定人类如何运用这把“普罗米修斯之火”。 === 结语：走进生命科学的殿堂 === 从细胞膜上的蛋白质泵到全球碳循环，从孟德尔的豌豆实验到人类基因组计划，生命科学始终在追问一个根本命题：'''生命为何如此运作'''？学习生物学，不仅是为了解生命的奥秘，更是为在技术爆炸的时代做出理性判断，在生态危机的十字路口选择前行方向。翻开这本教材，你将踏上一段穿越时空的旅程——从微观的分子世界到宏观的地球系统，从达尔文的加拉帕戈斯群岛到当代的合成生物学实验室，共同探索生命的本质与人类文明的未来。 pvsu6xpsf8tfgd1rcpc9smqxx99hyky 0 30862 181515 138786 2025-06-14T14:28:35Z Astroneering 39717 重写本节 181515 wikitext text/x-wiki 为什么要学习生物呢？ <br> 这似乎是个简单的问题，答案却像春天的蒲公英，轻轻一吹便散落四方。有人会说：“它像侦探小说一样引人入胜，每个细胞都在讲述生命的故事。”有人会答：“21世纪是生物的世纪，不学岂不是错过了时代的浪潮？”也有人直白：“高考要考，不学不行啊。”这些理由都像树叶的叶脉，各有各的走向，但终归指向同一个根系——生物学始终在我们身边，像空气般无声无息地渗透进生活的每个角落。你可以选择用好奇的眼睛凝视它，也可以把它当作不得不完成的任务，甚至避而远之……但无论你是否在意，它就在那里，不声不响地影响着我们的衣食住行。<br> 清晨刷牙时，牙膏中的酶制剂正在分解口腔细菌；午餐的面包发酵、酸奶制作，背后藏着微生物的代谢密码；甚至你此刻阅读时眼睛的调节功能，也依赖睫状肌与晶状体的精密配合。生物学从不遥远，它藏在呼吸的氧气里，在奔跑的肌肉中，在感冒时的免疫反应间。你可以不去深究，但若愿意多看一眼，便会发现：原来生命的奇迹，本就是属于每个人的日常。 <br> 或许有人觉得，生物学不过是课本上的冷知识，考试时背完就能抛诸脑后。可事实是，它远比考试重要得多。当医生告诉你“癌细胞逃逸了免疫监视”，当新闻播报“某地发现新物种”，当超市货架上出现“基因编辑食品”标签，这些时刻都需要一点生物学常识作为理解世界的工具。就像认识四季更替离不开地理知识，明白食物为何能提供能量，便绕不开细胞呼吸的原理。它不是抽象的名词，而是帮你拨开迷雾的透镜——透过它，你会明白为什么沙漠植物能活千年，为什么抗生素不能随便吃，为什么人类至今无法克隆出永生个体。<br> 生物学还藏着一种特别的思维方式。它教会我们观察：一片树叶的气孔分布，能解释植物如何在干旱中求生；它训练我们推理：DNA复制时的差错累积，如何导致癌症的发生；它提醒我们辩证：转基因技术既能拯救饥饿，也可能打破生态平衡。这种思维不会教条地告诉你“必须怎样”，而是像老朋友聊天那样，引导你去问“为什么”。当你开始追问“为什么沙漠动物会演化出夜行习性”，其实你已在用生态位理论思考适应性；当你质疑“基因编辑婴儿是否真的能定制天赋”，你已在用伦理框架审视科技边界。这种潜移默化的能力，远比记住“光合作用公式”更有价值。<br> 当然，生物学的意义不止于此。它像一扇窗，推开它，能看到人类文明的来路与去向。古代先民驯化野生稻，靠的是对遗传变异的朴素认知；现代科学家改造水稻基因，延续的却是同一条探索脉络。从达尔文在加拉帕戈斯群岛记录雀类喙形，到当代生物学家解析新冠病毒的刺突蛋白，贯穿始终的，是人类对生命本质的追问。而这份追问，最终会回到我们自身——我们从何而来？为何衰老？意识如何产生？这些问题的答案，或许就藏在你此刻身体里数万亿细胞的协作与冲突中。 <br> 所以，不必把生物学当作冷冰冰的学科。它更像是观察世界的放大镜，帮你读懂脚下的土壤、头顶的蓝天，甚至镜中那个自己。你可以带着功利心去学，为升学或职业铺路；也可以带着好奇心去学，为解开生命谜题而兴奋。即使暂时提不起兴趣，至少请记住：当你在公园散步、在厨房做饭、在医院就诊时，生物学从未缺席。它不是“冰美人”，只是需要一点耐心去靠近。翻开这本书，愿你能少些功利的背诵，多些探索的欣喜——毕竟，我们每个人都是这场生命史诗的亲历者。 15841eeajwia0wbf00zzsdihx04idve 0 30864 181517 139065 2025-06-14T14:41:12Z Astroneering 39717 重写本节 181517 wikitext text/x-wiki == 生命系统的层次：从微观到宏观的生命网络 == 当我们第一次接触到“生命系统”这个概念时，或许会感到些许陌生，但仔细想想，它其实与我们熟知的“生态系统”有着千丝万缕的联系。如果说生态系统是自然世界的一幅画作，那么生命系统就是构成这幅画的颜料——它不仅包括生态系统本身，还涵盖了从细胞到生物圈的每一个生命活动单元。你可以把生态系统想象成生命系统这棵大树上的一个枝条，而整棵树的根系、主干和所有枝叶，才构成了生物学研究的完整图景。 <br> 科学家们为了更清晰地描述生命的复杂性，将生命系统划分为九个层次：'''细胞、组织、器官、系统、个体、种群、群落、生态系统、生物圈'''。这个划分方式就像给生命活动套上不同倍数的放大镜——从微观的细胞器到宏观的地球生命网络，每个层次都对应着特定的研究视角。比如，当我们观察单个细胞的分裂过程时，聚焦的是最基础的生命单位；而当我们讨论全球气候变化对物种的影响时，目光则投向了整个生物圈的动态平衡。 <br> 以大熊猫为例，我们可以清晰地看到生命系统是如何逐级构建的。一只大熊猫的诞生始于一个受精卵细胞，这个细胞通过分裂与分化，逐渐形成肌肉组织、神经组织等不同功能的细胞群体。这些组织进一步组合成器官，比如心脏——它由肌肉组织负责收缩，神经组织调控节律，结缔组织提供支撑。多个器官协同工作，便构成了更高级的系统，如消化系统或循环系统。最终，这些系统共同协作，维持着大熊猫的生命活动。然而，生命系统的旅程并未止步于此。当一群大熊猫生活在同一片竹林中，它们就构成了'''种群'''；而竹林里所有的生物——包括竹子、昆虫、微生物——则组成了'''群落'''。更进一步，这些生物与阳光、水分、土壤等非生物环境相互作用，便形成了完整的'''生态系统'''。 <br> 但生命系统的奥秘远不止于此。有些生物的存在方式甚至打破了我们对“个体”的传统认知。例如大肠杆菌，它既是单个细胞，也是一个完整的生物个体；而当一只北极熊独自生活在冰原上时，它既是独立的个体，也是北极生态系统中的关键物种，甚至可能成为某个濒危种群的唯一幸存者。这种层级的重叠性，恰恰体现了生命系统的多样性与灵活性。 <br> 不过，不同生物的层级结构并非完全一致。植物学家在研究竹子时发现，植物个体的构建方式与动物截然不同。竹子并没有像人类那样明确的“循环系统”或“呼吸系统”，它的生命活动直接由根、茎、叶等器官完成。这种差异提醒我们：生命系统的层级划分不是机械的模板，而是服务于不同研究需求的分析工具。 <br> 理解生命系统的层次性，本质上是在构建一种观察世界的思维框架。当你凝视一片落叶时，看到的不仅是飘零的枯黄，更是细胞衰老、组织分解、养分回归生态系统的全过程；当你在动物园看到熊猫进食时，眼前上演的其实是消化系统运作、种群行为研究、生态系统能量流动的多重交响曲。这种从微观到宏观的思维切换能力，正是生物学赋予我们的独特视角。<br> === 值得注意的细节 === 1. '''植物的特殊性'''：虽然生命系统理论适用于所有生物，但植物个体的构建方式与动物存在本质差异。以竹子为例，它的生命活动由根、茎、叶等器官直接完成，无需像动物那样通过“系统”整合器官功能。 <br> 2. '''层级重叠现象'''：某些生物个体可能同时占据多个生命系统层次。例如，单细胞生物既是个体，也是细胞；而某个濒临灭绝的物种，可能同时代表个体、种群乃至整个群落的存续状态。 <br> 3. '''定义的动态性'''：生命系统的定义并非绝对固定。随着研究的深入，科学家可能需要重新界定某些层次的边界。比如，微生物组（如人体肠道菌群）是否应被纳入“系统”层次，目前仍是学术界讨论的热点。 <br> === 生命系统层次定义一览（供理解参考） === - 细胞：生命活动的基本单位，无论是草履虫的单细胞生存，还是人类细胞的复杂分工，都离不开这个最小结构的运作。 <br> - 组织：相似功能的细胞群与其分泌物的集合，就像肌肉组织的收缩功能，或植物输导组织的物质运输作用。 <br> - 器官：多种组织形成的有机整体，如人类的心脏或植物的叶片，各自承担特定的生理功能。 <br> - 系统：多个器官配合完成重大生命活动的功能网络，典型存在于动物界，如哺乳动物的神经系统。 <br> - 个体：能够独立完成生命活动的完整生物体，其范围可小至草履虫，大至蓝鲸。 <br> - 种群：同一区域同种生物的集合，如卧龙保护区内的所有大熊猫。 <br> - 群落：不同物种在特定环境中的生态网络，如热带雨林中植物、昆虫、鸟类的共生关系。 <br> - 生态系统：生物群落与其无机环境相互作用的动态平衡系统，例如一片湿地或整片海洋。 <br> - 生物圈：地球上所有生态系统的总和，是生命存在的最大尺度空间。 === 课后习题 === {{ExampleRobox|title=课后习题：}} 1. 如果以校园为研究对象，请尝试描述其中包含的生命系统层次（例如：教室里的绿植、操场上的蚂蚁群落、校园整体的生态特征）。<br> 2. 观察你身边的宠物猫或狗，试着分析它身上涉及哪些生命系统层次？如果换成一盆绿萝，这种分析会有何不同？ {{Robox/Close}} === 参考资料 === 1. 《普通高中课程标准实验教科书·生物必修一·分子与细胞》第一章<br> 2. 生命系统各层次定义参考《生态学名词》（第二版）及《现代生物学导论》 lcy9aatvm7ewaw61rq2xsm1fuuih4lz 2 30870 181523 140586 2025-06-14T17:55:27Z Aqurs1 50970 Aqurs1移動頁面[[User:唐舞麟]]至[[User:Astroneering]]：​当重命名用户“[[Special:CentralAuth/唐舞麟|唐舞麟]]”至“[[Special:CentralAuth/Astroneering|Astroneering]]”时自动移动页面 140586 wikitext text/x-wiki [[/沙盒]] =测试页 1= {{ExampleRobox|title=例題1}}宇蓁到便利商店買<math>2</math>個相同價錢的御飯糰和<math>1</math>瓶<math>42</math>元的優酪乳，在沒有任何促銷優惠下，總共花了<math>108</math>元。請問宇蓁買的御飯糰每個幾元？ {{Robox/Close}} =测试页 2= {{TextBox|1= ;Systems of linear equations are common in science and mathematics. These two examples from high school science {{harv|O'Nan|1990}} give a sense of how they arise.<br> 线性方程组在科学和数学中很常见。这两个来自《高中科学》（奥南1990）的例子让我们了解了它们是如何产生的。 {{anchor|physics problem}}The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances.<br> 第一个例子来自物理学。假设我们得到三个物体，一个质量已知为2公斤，要求出另外两个物体质量。进一步假设，使用天平进行试验可以产生这两种平衡。 <center> {|cellpadding=20px |[[Image:Linalg_balance_1.png|x130px]] |[[Image:Linalg_balance_2.png|x130px]] |} </center> Since the sum of magnitudes of the torques of the clockwise forces equal those of the counter clockwise forces (the torque of an object rotating about a fixed origin is the cross product of the force on it and its position vector relative to the origin; gravitational acceleration is uniform we can divide both sides by it). The two balances give this system of two equations.<br> 由于顺时针力的力矩大小之和等于逆时针力的大小（绕固定原点旋转的物体的力矩是其上的力与其相对于原点的位置矢量的叉积；重力加速度是均匀的，我们可以用它来划分两边）。这两个天平给出了这个由两个方程组成的方程组。 :<math>\begin{array}{rl} 40h+15c &= 100 \\ 25c &= 50+50h \end{array}</math> <quiz display=simple points=1/1> {Can you finish the solution?<br>你能求出解吗？ |type="{}" coef="2"} c = { 4_2 } kg h = { 1_2 } kg </quiz> {{anchor|chemistry problem}}The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene <math>\hbox{C}_7\hbox{H}_8</math> and nitric acid <math>\hbox{H}\hbox{N}\hbox{O}_3</math> to produce trinitrotoluene <math>\hbox{C}_7\hbox{H}_5\hbox{O}_6\hbox{N}_3</math> along with the byproduct water (conditions have to be controlled very well, indeed&mdash; trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction 线性系统的第二个例子来自化学。我们可以在受控制的情况下混合甲苯（<math>\hbox{C}_7\hbox{H}_8</math>） 和硝酸（<math>\hbox{H}\hbox{N}\hbox{O}_3</math>）的生产条件三硝基甲苯（<math>\hbox{C}_7\hbox{H}_5\hbox{O}_6\hbox{N}_3</math>）及其副产物水 （条件必须控制得很好，实际上，三硝基甲苯被称为TNT）。这些成分应该按多大比例混合？每种元素在反应前存在的原子数 :<math> x\,{\rm C}_7{\rm H}_8\ +\ y\,{\rm H}{\rm N}{\rm O}_3 \quad\longrightarrow\quad z\,{\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ w\,{\rm H}_2{\rm O} </math> must equal the number present afterward. Applying that principle to the elements C, H, N, and O in turn gives this system. 一定与反应后存在的原子数相等。根据这个反应式中的C、H、N和O元素的守恒。 :<math>\begin{array}{rl} 7x &= 7z \\ 8x +1y &= 5z+2w \\ 1y &= 3z \\ 3y &= 6z+1w \end{array}</math> <quiz display=simple points=1/1> {Can you balance the equation? 你能配平这个方程吗？ |type="{}" coef="4"} { 1_1 }<math>{\rm C}_7{\rm H}_8\ +\ </math>{ 3_1 }<math>{\rm H}{\rm N}{\rm O}_3\quad\longrightarrow\quad</math>{ 1_1 }<math>{\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ </math>{ 3_1 }<math>{\rm H}_2{\rm O}</math> </quiz> To finish each of these examples requires solving a system of equations. In each, the equations involve only the first power of the variables. This chapter shows how to solve any such system. 要完成这些例子中的每一个都需要解一个方程组。在每一个方程中，方程只涉及未知数的一次方。本章介绍如何求解任何此类方程组。 }} =测试页 3= [[高中生物/临时]]<br> =测试页 4= {{Expand language|1=en|time=2021-03-21T02:21:44+00:00}} {{noteTA |1=zh-cn:空气质量指数; zh-hk:空气质素指数; zh-tw:空气品质指数;}} [[File:California Fires MODIS081715.jpg|thumb|210px|2015年8月从太空中看[[加州]]野火与浓烟，造成空气质量指数变差，影响居民健康。]] '''空气品质指标'''<ref>{{Cite web|title=行政院环境保护署 - 空气品质监测网|url=https://airtw.epa.gov.tw/|accessdate=2021-03-19|last=行政院环境保护署|work=airtw.epa.gov.tw|language=zh-TW}}</ref>（{{lang-en|'''A'''ir '''Q'''uality '''I'''ndex, '''AQI'''}}）、'''空气质量指数'''是定量描述空气质量状况的[[非线性]][[无量纲量|无量纲]]指数。其数值越大、等级和类别越高、颜色越深，代表空气污染状况越严重，对人体的健康危害也就越大。<br /> 由于[[颗粒物]]没有小时浓度标淮，基于24小时平均浓度计算的AQI相对于空气质量的小时变化会存在一定的滞后性，因此，当首要污染物为PM_2.5和PM₁₀时，在看AQI的同时还要兼顾其实时浓度数据。相关单位为弥补滞后性，同时发布了“即时空气质量指数”，所有污染物均采用当前1小时平均浓度计算。要注意“实时空气质量指数”不是AQI。<ref>{{Cite web |url=http://www.semc.gov.cn/aqi/home/Detail.aspx?id=656a350f-e319-4cf5-9da8-a516be578c5a |title=本次实时发布系统改版主要做了哪些修改？ |date=2014-04-14 |publisher=上海市环境监测中心 |language=zh-cn |accessdate=2014-04-19 |archive-url=https://web.archive.org/web/20140419125705/http://www.semc.gov.cn/aqi/home/Detail.aspx?id=656a350f-e319-4cf5-9da8-a516be578c5a |archive-date=2014-04-19 |dead-url=yes }}</ref><br /> 需要说明的是，AQI的计算结果很大程度上取决于相应地区空气质量分指数及对应的污染物项目浓度指数表，最终的计算结果需要参考相应的浓度指数表才具有实际意义。 对于中国，AQI与原来发布的空气污染指数（API）有着很大的区别。AQI分级计算参考的标淮是GB 3095-2012《环境空气质量标淮》（现行），参与评价的污染物为[[二氧化硫|SO₂]]、[[二氧化氮|NO₂]]、[[PM10|PM₁₀]]、[[PM2.5|PM_2.5]]、[[臭氧|O₃]]、[[一氧化碳|CO]]等六项，每小时发布一次；而API分级计算参考的标淮是GB 3095-1996《环境空气质量标淮》（已作废），评价的污染物仅为SO₂、NO₂和PM₁₀等三项，每天发布一次。因此，AQI采用的标淮更严、污染物指标更多、发布频次更高，其评价结果也将更加接近公众的真实感受。 == 计算方法 == === 空气质量分指数 === 对照各项污染物的分级浓度限值，以[[细颗粒物]]（PM_2.5）、[[可吸入颗粒物]]（PM₁₀）、[[二氧化硫]]（SO₂）、[[二氧化氮]]（NO₂）、[[臭氧]]（O₃）、[[一氧化碳]]（CO）等各项污染物的实测浓度值（其中PM_2.5、PM₁₀为24小时平均浓度）分别计算得出'''空气质量分指数'''（{{lang-en|'''I'''ndividual '''A'''ir '''Q'''uality '''I'''ndex, '''IAQI'''}}）。 <div style="text-align: center;">{{math|big=1|IAQI_P {{=}} {{sfrac|IAQI_H''i'' - IAQI_L''o''|BP_H''i'' - BP_L''o''}} (''C''_P - BP_L''o'') + IAQI_L''o''}}</div> 式中： :{{math|IAQI_P}}——相应地区的污染物项目{{math|P}}的空气质量分指数； :{{math|''C''_P}}——相应地区的污染物项目{{math|P}}的质量浓度值； :{{math|BP_H''i''}}——空气质量分指数对应的污染物项目浓度限值表中与{{math|''C''_P}}相近的污染物浓度限值的高位值； :{{math|BP_L''o''}}——空气质量分指数对应的污染物项目浓度限值表中与{{math|''C''_P}}相近的污染物浓度限值的低位值； :{{math|IAQI_H''i''}}——空气质量分指数对应的污染物项目浓度限值表中与{{math|BP_H''i''}}对应的空气质量分指数； :{{math|IAQI_L''o''}}——空气质量分指数对应的污染物项目浓度限值表中与{{math|BP_L''o''}}对应的空气质量分指数。 === 空气质量指数 === <div style="text-align: center;">{{math|big=1|AQI {{=}} max {IAQI₁, IAQI₂, IAQI₃,…, IAQI_''n''} }}</div> 式中： :{{math|IAQI}}——空气质量分指数； :{{math|''n''}}——污染物项目。 简单来说，AQI就是在各IAQI中取最大值。<br> AQI大于50时，IAQI最大的污染物为首要污染物。若IAQI最大的污染物为两项或两项以上时，并列为首要污染物。IAQI大于100的污染物为超标污染物。 == 空气质量分指数分级 == === 中华人民共和国 === 中华人民共和国生态环境部科技标淮司于2012年发布了《环境空气质量指数（AQI）技术规定（试行）（HJ 633—2012）》，于2016年1月1日起实施<ref>{{cite web |author1=中华人民共和国环境保护部 |title=环境空气质量指数(AQI)技术规定 (试行) |url=http://kjs.mee.gov.cn/hjbhbz/bzwb/jcffbz/201203/W020120410332725219541.pdf |accessdate=2019-09-10 |archive-url=https://web.archive.org/web/20190713234941/http://kjs.mee.gov.cn/hjbhbz/bzwb/jcffbz/201203/W020120410332725219541.pdf |archive-date=2019-07-13 |dead-url=yes }}</ref>。 {| class="wikitable" |+ 空气质量分指数及对应污染物项目浓度限值 |- ! 空气质量分指数 !! colspan=10|污染物浓度（μg/m³）,其中CO单位为（mg/m³） |- !IAQI !! 二氧化硫(SO₂)24小时平均值 !! 二氧化硫(SO₂)1小时平均值^（1） !! 二氧化氮(NO₂)24小时平均值 !! 二氧化氮(NO₂)1小时平均值^（1） !! 颗粒物(粒径小于10μm)24小时平均值 !! 一氧化碳（CO）24小时平均值 !! 一氧化碳（CO）1小时平均值^（1） !! 臭氧（O₃）1小时平均值 !! 臭氧（O₃）8小时滑动平均值 !! 颗粒物(粒径小于2.5μm)24小时平均值 |- ||0||0||0||0||0||0||0||0||0||0||0 |- ||50||50||150||40||100||50||2||5||160||100||35 |- ||100||150||500||80||200||150||4||10||200||160||75 |- ||150||475||650||180||700||250||14||35||300||215||115 |- ||200||800||800||280||1200||350||24||60||400||265||150 |- ||300||1600||^（2）||565||2340||420||36||90||800||800||250 |- ||400||2100||^（2）||750||3090||500||48||120||1000||⁽³⁾||350 |- ||500||2620||^（2）||940||3840||600||60||150||1200||⁽³⁾||500 |- ||说明|| colspan=10| ⁽¹⁾二氧化硫(SO₂)、二氧化氮(NO₂)和一氧化碳(CO)的1小时平均浓度限值仅用于实时报，在日报中需使用相应污染物的24小时平均浓度限值。 ⁽²⁾二氧化硫(SO₂)1小时平均浓度值高于800μg/m³的，不再进行其空气质量分指数计算，二氧化硫(SO₂)空气质量分指数按24小时平均浓度计算的分指数报告。 ⁽³⁾臭氧(O₃)8小时平均浓度值高于800μg/m³的，不再进行其空气质量分指数计算，臭氧(O₃)空气质量分指数按1小时平均浓度计算的分指数报告。 |} {| class="wikitable" |+ 空气质量指数及相关信息 |- ! AQI数值 !! AQI级别 !! colspan=2|AQI类别及表示颜色!!对健康的影响!!建议采取的措施 |- | 0～50 ||一级 || 优||style="background:#00e400;"| 绿色||空气质量令人满意，基本无空气污染||各类人群可正常活动 |- | 51～100 || 二级 || 良 ||style="background:#ff0;"|黄色 || 空气质量可接受，但某些污染物可能对极少数异常敏感人群健康有较弱影响 ||极少数异常敏感人群应减少户外活动 |- | 101～150 || 三级 || 轻度污染 ||style="background:#ff7e00;"|橙色 || 易感人群症状有轻度加剧，健康人群出现刺激症状 ||儿童、老年人及心脏病、呼吸系统疾病患者应减少长时间、高强度的户外锻炼 |- | 151～200 ||四级 || 中度污染||style="background:#f00;"|红色 || 进一步加剧易感人群症状，可能对健康人群心脏、呼吸系统有影响 ||儿童、老年人及心脏病、呼吸系统疾病患者应避免长时间、高强度的户外锻炼，一般人群适量减少户外运动 |- | 201～300 || 五级 || 重度污染 ||style="background:#99004c;"|紫色 || 心脏病和肺病患者症状显著加剧，运动耐受力降低，健康人群普遍出现症状 ||儿童、老年人及心脏病、呼吸系统疾病患者应停留在室内，停止户外活动，一般人群应避免户外活动 |- | >300 || 六级 || 严重污染||style="background:#7e0023;"| 褐红色 || 健康人群运动耐受力降低，有明显强烈症状，提前出现某些疾病 ||儿童、老年人及心脏病、呼吸系统疾病患者应停留在室内，避免体力消耗，一般人群应避免户外活动 |} === 美国 === 美国环境保护署（EPA）已开发出用于报告空气品质的空气品质指数（AQI）。该空气品质指数分为六类，显示健康影响程度的高低。空气品质指数值超过300代表有害的空气品质，低于50空气品质良好。 [[Image:Pm25-24a-super.gif|thumb|210px|PM_2.5 24小时空气品质指数(AQI)循环，美国环保署提供]] {| class="wikitable" |+美国空气分指数及对应的污染物项目浓度指数表<ref>{{cite web |title=AirNow |url=https://www.airnow.gov/publications/air-quality-index/technical-assistance-document-for-reporting-the-daily-aqi/ |website=Technical Assistance Document for the Reporting of Daily Air Quality |accessdate=2020-05-28}}</ref> |- ! AQI指数 !! [[臭氧]]（O₃）1小时平均/（μg/m³ !! [[臭氧]]（O₃）8小时滑动平均/（μg/m³） !! [[细颗粒物]]PM_2.5（粒径小于等于2.5μm）24小时平均/（μg/m³） !! [[可吸入颗粒物]]PM₁₀（粒径小于等于10μm）24小时平均/（μg/m³） !! [[一氧化碳]]（CO）8小时平均/（mg/m³）!! [[二氧化硫]]（SO₂）1小时平均/（μg/m³） ![[二氧化硫]]（SO₂）24小时平均/（μg/m³） ![[二氧化氮]]（NO₂）1小时平均/（μg/m³） |- | 0 || - || 0 || 0 || 0 || 0 || 0 | - |0 |- | 50|| - || 108 || 12|| 54|| 5.038|| 91.7 | - |99.64 |- | 100|| 250 || 140 || 35.4|| 154|| 10.763|| 196.5 | - |188 |- | 150|| 328|| 170|| 55.4|| 254|| 14.198|| 484.7 | - |676.8 |- | 200|| 408|| 210|| 150.4|| 354|| 17.633|| 793 |799.1 |1220 |- | 300|| 808 || 400|| 250.4|| 424|| 34.35|| - |1582.5 |2350 |- |400 |1008 | - |350.4 |504 |46.258 | - |2106.5 |3100 |- | 500|| 1208|| -|| 500.4|| 600|| 57.708||- |2630.5 |3850 |} ==== 美国空气品质指数颜色分级 ==== {| |- | {| border="2" cellspacing="0" style="width:480px; float:left;" |- style="vertical-align:top; background:#036;" | style="width:31%;"| <strong><span style="color:#fff; font-family:arial; font-size:1em;">空气品质指数（AQI）</span></strong>|| style="background:#036; width:32%;"|<span style="color:#fff; font-family:arial; font-size:1em;">'''健康令人担忧的程度'''</span>|| style="background:#036; width:37%;"|<span style="color:#fff; font-family:arial; font-size:1em;">'''颜色'''</span> |- style="vertical-align:top; background:#00e400;" | style="width:31%;"| 0 to 50|| style="width:32%;"|好|| style="width:37%;"|绿色 |- style="vertical-align:top; background:#ff0;" | style="width:31%;"| 51 to 100|| style="width:32%;"|中等|| style="width:37%;"|黄色 |- style="vertical-align:top; background:#ff7e00;" | style="width:31%;"| 101 to 150|| style="width:32%;"|不适于敏感人群|| style="width:37%;"|橘色 |- style="vertical-align:top; background:#f00;" | style="width:31%;"| <span style="color:#fff;">151 to 200</span>|| style="width:32%;"|<span style="color:#fff;">不健康</span>|| style="width:37%;"|<span style="color:#fff;">红色</span> |- style="vertical-align:top; background:#99004c;" | style="width:31%;"| <span style="color:#fff;">201 to 300</span>|| style="width:32%;"|<span style="color:#fff;">非常不健康</span>|| style="width:37%;"|<span style="color:#fff;">紫色</span> |- style="vertical-align:top; background:#7e0023;" | style="width:31%;"| <span style="color:#fff;">301 to 500</span>|| style="width:32%;"|<span style="color:#fff;">危险</span>|| style="width:37%;"|<span style="color:#fff;">枣红色</span> |} |} === 英国 === 在英国最常用的空气品质指数是每日空气品质指数，由空气污染物对医疗健康影响委员会(COMEAP)所推荐。这个指数设有十个点，更进一步分成4个等级：低，中，高和非常高。每个等级都有针对高危险群和一般人的健康建议。 {| class="wikitable" |- ! 空气污染 !! 值!! 对有危险个人的健康资讯!! 对于一般人群的健康资讯 |- |style="color:#009900;font-weight:bold"| 低 ||style="color:#009900;font-weight:bold"| 1–3|| 享受你平时的户外活动。|| 享受你平时的户外活动。 |- |style="color:#FF9900;font-weight:bold"| 中度||style="color:#FF9900;font-weight:bold"| 4–6|| 成人和儿童肺部有问题，和成人有心脏问题，而且有症状，应考虑减少剧烈的体力活动，尤其是在户外。 || 享受你平时的户外活动。 |- |style="color:#FF0000;font-weight:bold"| 高||style="color:#FF0000;font-weight:bold"| 7–9|| 成人和儿童肺部有问题，和成人心脏有问题，应减少剧烈的体力劳动尤其是在户外，特别是如果他们有症状。有哮喘的人可能会发现他们需要更频繁地需要药物吸入。年纪大的人也应减少体力消耗。 || 任何人感到不适，如眼痛，咳嗽或喉咙痛应该考虑减少活动，尤其是在户外。 |- |style="color:#990099;font-weight:bold"| 非常高||style="color:#990099;font-weight:bold" | 10|| 成人和儿童肺的问题，有心脏问题的成人，老年人，应避免剧烈的体力活动。有哮喘的人可能会发现他们需要更频繁地需要药物吸入。 || 减少体力消耗，特别是在户外，尤其是如果你遇到的症状，如咳嗽或喉咙痛。 |} {| class="wikitable" |- ! 指数 !! 臭氧，运行8小时平均值 (微克/米³) !! 二氧化氮，每小时平均 (微克/米³) !! 二氧化硫15分钟平均值 (微克/米³) !! PM2.5颗粒，24小时平均值 (微克/米³) !! PM10颗粒，24小时平均值 (微克/米³) |-style="background-color:#9CFF9C; color:#444;font-weight:bold" | 1|| 0-33 || 0-67 || 0-88 || 0-11 || 0-16 |-style="background-color:#31FF00;color:#333;font-weight:bold" | 2|| 34-66 || 68-134|| 89-177 || 12-23 || 17-33 |-style="background-color:#31CF00;color:#222;font-weight:bold" | 3|| 67-100 || 135-200 || 178-266 || 24-35 || 34-50 |-style="background-color:#FFFF00;color:#666;font-weight:bold" | 4|| 101-120 || 201-267 || 267-354 || 36-41 || 51-58 |-style="background-color:#FFCF00;color:#666;font-weight:bold" | 5|| 121-140 || 268-334|| 355-443 || 42-47 || 59-66 |-style="background-color:#FF9A00;color:#222;font-weight:bold" | 6|| 141-160 || 335-400|| 444-532 || 48-53 || 67-75 |-style="background-color:#FF6464;color:#000;font-weight:bold" | 7|| 161-187 || 401-467|| 533-710 || 54-58 || 76-83 |-style="background-color:#FF0000;color:#080805;font-weight:bold" | 8|| 188-213 || 468-534|| 711-887 || 59-64 || 84-91 |-style="background-color:#990000;color:#fff;font-weight:bold" | 9|| 214-240 || 535-600|| 888-1064 || 65-70 || 92-100 |-style="background-color:#CE30FF;color:#FFF;font-weight:bold" | 10|| ≥ 241 || ≥ 601 || ≥ 1065 || ≥ 71 || ≥ 101 |} == 相关 == * [[空气污染指数]] * [[台湾空气品质指数]] * [[澳门空气质量指数]] * [[中国空气质素指数]] * [[香港空气质素健康指数]] ==外部连结== # [https://aqicn.org/map/hk/ 亚洲空气污染：实时空气质量指数地图] # [https://web.archive.org/web/20130116100627/http://datacenter.mep.gov.cn/ 中华人民共和国环境保护部--数据中心] == 参考 == {{reflist}} {{自然资源}} {{Authority control}} [[Category:空气质量指数| ]] =测试页 5= =测试页 6= ktbtiaz8pd1p42hlo6v1v7hh5dlvj89 0 30876 181513 174310 2025-06-14T14:12:12Z Astroneering 39717 重写本节 181513 wikitext text/x-wiki == 生物体内的无机物 == === 水的分类与生物学功能 === 水是生物体内含量最丰富的化合物，也是生命活动不可或缺的介质。其分子由两个氢原子和一个氧原子构成（H₂O），具有极性特性，能够形成氢键，从而表现出高比热容、高汽化热以及优异的溶剂能力等物理化学性质。在生物体内，水以两种形式存在：'''自由'''水与'''结合水'''。<br> '''自由水'''是指未与其他物质结合、可自由流动的水分子，占细胞总水量的绝大部分。自由水直接参与多种生化反应，是细胞内化学反应的核心介质。例如，在动物消化系统中，水作为溶剂促进消化酶对营养物质的水解作用；在植物光合作用中，水不仅是光反应的原料（参与光系统II的水裂解反应），还通过蒸腾作用驱动水分和矿质离子的长距离运输。此外，自由水在维持细胞渗透压平衡中起关键作用：血液中的血浆以水为基质，负责运输氧气、营养物质及代谢废物；淋巴液和组织液中的水则构成细胞外液，为细胞提供稳定的内环境。水的高汽化热特性使其在温度调节中具有重要意义。例如，动物通过汗液蒸发散热维持体温稳态，植物通过蒸腾作用降低叶片温度以保护光合机构。<br> '''结合水'''则与蛋白质、多糖等大分子通过氢键结合，成为细胞结构的一部分。这类水分子失去流动性，但仍对细胞结构的稳定性至关重要。例如，植物细胞壁中的结合水与纤维素形成氢键网络，维持细胞壁的机械强度；动物细胞中的结合水则参与细胞膜双分子层的稳定化。自由水与结合水的比值动态平衡直接影响生物体的代谢强度与抗逆性：比值较高时（自由水占优），代谢活跃但抗逆性较弱；比值较低时（结合水占优），代谢减缓但抗逆性增强。这一特性在种子萌发和休眠过程中尤为显著——种子在休眠时结合水比例上升，代谢速率降低，从而增强对干旱、低温等逆境的耐受性；萌发时自由水比例恢复，代谢活动迅速激活。 === 无机盐的存在形式与生理意义 === 无机盐通常以离子形式存在于生物体内，其功能涉及多个层面。常见的阳离子包括钠（Na⁺）、钾（K⁺）、钙（Ca²⁺）、镁（Mg²⁺）等，阴离子则主要包括氯（Cl⁻）、磷酸根（PO₄³⁻）、碳酸氢根（HCO₃⁻）等。这些离子通过跨膜转运蛋白（如钠钾泵）维持细胞内外液的离子浓度梯度，进而调控细胞电位、渗透压及酸碱平衡。<br> 在神经与肌肉活动中，无机盐的作用尤为关键。钠离子（Na⁺）和钾离子（K⁺）通过钠钾泵的主动运输机制建立跨膜电位，为动作电位的产生与传导提供基础。钙离子（Ca²⁺）作为细胞内第二信使，参与肌肉收缩、神经递质释放及基因表达调控。例如，骨骼肌中肌质网释放的Ca²⁺与肌钙蛋白结合，触发肌丝滑行，最终导致肌肉收缩。此外，无机盐通过缓冲体系维持内环境稳态。碳酸氢盐缓冲体系（HCO₃⁻/CO₂）是血液中最主要的pH调节系统，通过肺和肾脏的协同作用，将pH稳定在7.35-7.45的狭窄范围内。<br> 无机盐对酶活性的调节亦至关重要。许多酶需要金属离子作为辅因子，例如叶绿素的核心结构含镁离子（Mg²⁺），直接影响光能吸收效率；乙醇脱氢酶依赖锌离子（Zn²⁺）催化酒精氧化反应。此外，三磷酸腺苷（ATP）的水解反应需要Mg²⁺作为辅助因子，以中和ATP的负电荷，促进酶促反应进行。<br> === 实验验证与实例分析 === 通过实验可直观验证水与无机盐的功能。将小麦种子晾晒后，自由水大量流失，细胞代谢速率显著下降；进一步高温焚烧后，结合水与有机物被分解，残留的灰烬即为无机盐（如磷酸钙、氯化钠等）。这一过程揭示了水和无机盐在生物体中的存在形式及其对代谢的支撑作用。<br> 无机盐缺乏症的表现进一步印证了其生理意义。植物缺镁会导致叶绿素合成障碍，表现为叶片黄化（缺绿症）；缺磷则影响ATP合成，抑制能量代谢。在人体中，钙离子（Ca²⁺）不足会引发骨质疏松，而铁（Fe²⁺）缺乏导致血红蛋白合成障碍，引发缺铁性贫血。此外，钠钾泵功能紊乱可能引发电解质失衡，表现为肌肉抽搐或心律失常，凸显了无机盐在维持细胞电生理特性中的核心地位。<br> === 课堂小结 === 水与无机盐共同构成生物体的基础化学环境。自由水与结合水的动态平衡调控代谢强度，而无机盐则通过离子形式维系生理稳态。两者的协同作用确保生物体在复杂环境中维持正常生命活动，从分子水平的酶促反应到宏观层面的代谢调控，均离不开无机物的支持。 ridmpbx2ze1o1qftsxcnjgwded1l55s 181526 181513 2025-06-15T04:10:14Z 103.232.212.91 /* 水的分类与生物学功能 */小修改 181526 wikitext text/x-wiki == 生物体内的无机物 == === 水的分类与生物学功能 === 水是生物体内含量最丰富的化合物，也是生命活动不可或缺的介质。其分子由两个氢原子和一个氧原子构成（H₂O），具有极性特性，能够形成氢键，从而表现出高比热容、高汽化热以及优异的溶剂能力等物理化学性质。在生物体内，水以两种形式存在：'''自由水'''与'''结合水'''。<br> '''自由水'''是指未与其他物质结合、可自由流动的水分子，占细胞总水量的绝大部分。自由水直接参与多种生化反应，是细胞内化学反应的核心介质。例如，在动物消化系统中，水作为溶剂促进消化酶对营养物质的水解作用；在植物光合作用中，水不仅是光反应的原料（参与光系统II的水裂解反应），还通过蒸腾作用驱动水分和矿质离子的长距离运输。此外，自由水在维持细胞渗透压平衡中起关键作用：血液中的血浆以水为基质，负责运输氧气、营养物质及代谢废物；淋巴液和组织液中的水则构成细胞外液，为细胞提供稳定的内环境。水的高汽化热特性使其在温度调节中具有重要意义。例如，动物通过汗液蒸发散热维持体温稳态，植物通过蒸腾作用降低叶片温度以保护光合机构。<br> '''结合水'''则与蛋白质、多糖等大分子通过氢键结合，成为细胞结构的一部分。这类水分子失去流动性，但仍对细胞结构的稳定性至关重要。例如，植物细胞壁中的结合水与纤维素形成氢键网络，维持细胞壁的机械强度；动物细胞中的结合水则参与细胞膜双分子层的稳定化。自由水与结合水的比值动态平衡直接影响生物体的代谢强度与抗逆性：比值较高时（自由水占优），代谢活跃但抗逆性较弱；比值较低时（结合水占优），代谢减缓但抗逆性增强。这一特性在种子萌发和休眠过程中尤为显著——种子在休眠时结合水比例上升，代谢速率降低，从而增强对干旱、低温等逆境的耐受性；萌发时自由水比例恢复，代谢活动迅速激活。 === 无机盐的存在形式与生理意义 === 无机盐通常以离子形式存在于生物体内，其功能涉及多个层面。常见的阳离子包括钠（Na⁺）、钾（K⁺）、钙（Ca²⁺）、镁（Mg²⁺）等，阴离子则主要包括氯（Cl⁻）、磷酸根（PO₄³⁻）、碳酸氢根（HCO₃⁻）等。这些离子通过跨膜转运蛋白（如钠钾泵）维持细胞内外液的离子浓度梯度，进而调控细胞电位、渗透压及酸碱平衡。<br> 在神经与肌肉活动中，无机盐的作用尤为关键。钠离子（Na⁺）和钾离子（K⁺）通过钠钾泵的主动运输机制建立跨膜电位，为动作电位的产生与传导提供基础。钙离子（Ca²⁺）作为细胞内第二信使，参与肌肉收缩、神经递质释放及基因表达调控。例如，骨骼肌中肌质网释放的Ca²⁺与肌钙蛋白结合，触发肌丝滑行，最终导致肌肉收缩。此外，无机盐通过缓冲体系维持内环境稳态。碳酸氢盐缓冲体系（HCO₃⁻/CO₂）是血液中最主要的pH调节系统，通过肺和肾脏的协同作用，将pH稳定在7.35-7.45的狭窄范围内。<br> 无机盐对酶活性的调节亦至关重要。许多酶需要金属离子作为辅因子，例如叶绿素的核心结构含镁离子（Mg²⁺），直接影响光能吸收效率；乙醇脱氢酶依赖锌离子（Zn²⁺）催化酒精氧化反应。此外，三磷酸腺苷（ATP）的水解反应需要Mg²⁺作为辅助因子，以中和ATP的负电荷，促进酶促反应进行。<br> === 实验验证与实例分析 === 通过实验可直观验证水与无机盐的功能。将小麦种子晾晒后，自由水大量流失，细胞代谢速率显著下降；进一步高温焚烧后，结合水与有机物被分解，残留的灰烬即为无机盐（如磷酸钙、氯化钠等）。这一过程揭示了水和无机盐在生物体中的存在形式及其对代谢的支撑作用。<br> 无机盐缺乏症的表现进一步印证了其生理意义。植物缺镁会导致叶绿素合成障碍，表现为叶片黄化（缺绿症）；缺磷则影响ATP合成，抑制能量代谢。在人体中，钙离子（Ca²⁺）不足会引发骨质疏松，而铁（Fe²⁺）缺乏导致血红蛋白合成障碍，引发缺铁性贫血。此外，钠钾泵功能紊乱可能引发电解质失衡，表现为肌肉抽搐或心律失常，凸显了无机盐在维持细胞电生理特性中的核心地位。<br> === 课堂小结 === 水与无机盐共同构成生物体的基础化学环境。自由水与结合水的动态平衡调控代谢强度，而无机盐则通过离子形式维系生理稳态。两者的协同作用确保生物体在复杂环境中维持正常生命活动，从分子水平的酶促反应到宏观层面的代谢调控，均离不开无机物的支持。 45499fadgs3z7iceor38k5t82b0m15x 2 30899 181521 139378 2025-06-14T17:55:25Z Aqurs1 50970 Aqurs1移動頁面[[User:唐舞麟/沙盒]]至[[User:Astroneering/沙盒]]：​当重命名用户“[[Special:CentralAuth/唐舞麟|唐舞麟]]”至“[[Special:CentralAuth/Astroneering|Astroneering]]”时自动移动页面 139378 wikitext text/x-wiki =线性代数（来源于英文维基教科书）= __NOTOC__ {{book title|''Linear Algebra 线性代数''|An Introduction to Mathematical Discourse 数学话语导论}} <!-- removed per talk discussion after 7 mos., 31 Jul 2013 {{Outdated}} --> {{ambox |type=notice |style=float:right;margin-left:10px;margin-right:0px; |textstyle=padding:5px;font-size:120%;line-height:120%; |small=left |text=This book requires that you are familiar with calculus. This subject is covered by the wikibook Calculus.<br> 这本书要求你熟悉微积分。维基教科书《微积分》涵盖了这个主题。}} The book was designed specifically for students who had not previously been exposed to mathematics as mathematicians view it. That is, as a subject whose goal is to ''rigorously'' prove theorems starting from clear consistent definitions. This book attempts to build students up from a background where mathematics is simply a tool that provides useful calculations to the point where the students have a grasp of the clear and precise nature of mathematics. A more detailed discussion of the prerequisites and goal of this book is given in the introduction.<br> 这本书是专门为那些以前没有接触过数学的学生设计的，因为他们是数学家。也就是说，作为一个目标是从清晰一致的定义开始严格证明定理的学科。这本书试图建立学生从一个背景，数学只是一个工具，提供有用的计算点，学生有一个清晰和精确的数学性质的掌握。引言中对本书的先决条件和目标进行了更详细的讨论。 == Table of Contents == {{Algebra}} *[[Linear Algebra/Cover|Cover]] *[[Linear Algebra/Notation|Notation]] *[[Linear Algebra/Introduction|Introduction]] === Linear Systems === <ol type=I> <li>[[Linear Algebra/Solving Linear Systems|Solving Linear Systems]]{{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Gauss' Method|Gauss' Method]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Describing the Solution Set|Describing the Solution Set]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/General = Particular + Homogeneous|General = Particular + Homogeneous]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Comparing Set Descriptions|Comparing Set Descriptions]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Automation|Automation]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Linear Geometry of n-Space|Linear Geometry of ''n''-Space]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Vectors in Space|Vectors in Space]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Length and Angle Measures|Length and Angle Measures]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Reduced Echelon Form|Reduced Echelon Form]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Gauss-Jordan Reduction|Gauss-Jordan Reduction]] {{stage|100%|Jul 13, 2009}} #[[Linear Algebra/Row Equivalence|Row Equivalence]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Topic: Computer Algebra Systems|Topic: Computer Algebra Systems]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Topic: Input-Output Analysis|Topic: Input-Output Analysis]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Input-Output Analysis M File|Input-Output Analysis M File]] {{stage|100%|Mar 24 2008}} <li>[[Linear Algebra/Topic: Accuracy of Computations|Topic: Accuracy of Computations]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Topic: Analyzing Networks|Topic: Analyzing Networks]] {{stage|100%|Jul 13, 2009}} <li>[[Linear Algebra/Topic: Speed of Gauss' Method|Topic: Speed of Gauss' Method]] {{stage|50%|Mar 24, 2008}} </ol> === [[Linear Algebra/Vector Spaces|Vector Spaces]] {{stage|100%|Apr 17, 2009}} === <ol type=I> <li>[[Linear Algebra/Definition of Vector Space|Definition of Vector Space]]{{stage|100%|Apr 17, 2009}} #[[Linear Algebra/Definition and Examples of Vector Spaces|Definition and Examples]]{{stage|100%|Jun 18, 2009}} #[[Linear Algebra/Subspaces and Spanning sets|Subspaces and Spanning sets]]{{stage|100%|Jun 18, 2009}} <li>[[Linear Algebra/Linear Independence|Linear Independence]]{{stage|100%|Apr 17, 2009}} #[[Linear Algebra/Definition and Examples of Linear Independence|Definition and Examples]]{{stage|100%|Apr 17, 2009}} <li>[[Linear Algebra/Basis and Dimension|Basis and Dimension]]{{stage|100%|Apr 17, 2009}} #[[Linear Algebra/Basis|Basis]]{{stage|100%|Jun 18, 2009}} #[[Linear Algebra/Dimension|Dimension]]{{stage|100%|Apr 17, 2009}} #[[Linear Algebra/Vector Spaces and Linear Systems|Vector Spaces and Linear Systems]]{{stage|100%|Apr 17, 2009}} #[[Linear Algebra/Combining Subspaces|Combining Subspaces]]{{stage|100%|Apr 17, 2009}} <li>[[Linear Algebra/Topic: Fields|Topic: Fields]]{{stage|100%|Apr 17, 2009}} <li>[[Linear Algebra/Topic: Crystals|Topic: Crystals]]{{stage|100%|Apr 17, 2009}} <li>[[Linear Algebra/Topic: Voting Paradoxes|Topic: Voting Paradoxes]]{{stage|100%|Apr 17, 2009}} <li>[[Linear Algebra/Topic: Dimensional Analysis|Topic: Dimensional Analysis]]{{stage|100%|Apr 17, 2009}} </ol> === Maps Between Spaces === <ol type="I"> <li>[[Linear Algebra/Isomorphisms|Isomorphisms]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Definition and Examples of Isomorphisms|Definition and Examples]]{{stage|100%|July 19, 2009}} #[[Linear Algebra/Dimension Characterizes Isomorphism|Dimension Characterizes Isomorphism]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Homomorphisms|Homomorphisms]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Definition of Homomorphism|Definition of Homomorphism]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Rangespace and Nullspace|Rangespace and Nullspace]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Computing Linear Maps|Computing Linear Maps]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Representing Linear Maps with Matrices|Representing Linear Maps with Matrices]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Any Matrix Represents a Linear Map|Any Matrix Represents a Linear Map]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Matrix Operations|Matrix Operations]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Sums and Scalar Products|Sums and Scalar Products]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Matrix Multiplication|Matrix Multiplication]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Mechanics of Matrix Multiplication|Mechanics of Matrix Multiplication]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Inverses|Inverses]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Change of Basis|Change of Basis]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Changing Representations of Vectors|Changing Representations of Vectors]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Changing Map Representations|Changing Map Representations]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Projection|Projection]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Orthogonal Projection Onto a Line|Orthogonal Projection Onto a Line]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Gram-Schmidt Orthogonalization|Gram-Schmidt Orthogonalization]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Projection Onto a Subspace|Projection Onto a Subspace]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Line of Best Fit|Topic: Line of Best Fit]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Geometry of Linear Maps|Topic: Geometry of Linear Maps]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Markov Chains|Topic: Markov Chains]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Orthonormal Matrices|Topic: Orthonormal Matrices]]{{stage|100%|Jun 21, 2009}} </ol> === [[Linear Algebra/Determinants|Determinants]]{{stage|100%|Jun 21, 2009}} === <ol type="I"> <li>[[Linear Algebra/Definition of Determinant|Definition]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Exploration|Exploration]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Properties of Determinants|Properties of Determinants]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/The Permutation Expansion|The Permutation Expansion]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Determinants Exist|Determinants Exist]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Geometry of Determinants|Geometry of Determinants]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Determinants as Size Functions|Determinants as Size Functions]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Other Formulas for Determinants|Other Formulas for Determinants]]{{stage|100%|Jun 21, 2009}} #[[Linear Algebra/Laplace's Expansion|Laplace's Expansion]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Cramer's Rule|Topic: Cramer's Rule]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Speed of Calculating Determinants|Topic: Speed of Calculating Determinants]]{{stage|100%|Jun 21, 2009}} <li>[[Linear Algebra/Topic: Projective Geometry|Topic: Projective Geometry]]{{stage|100%|Jun 21, 2009}} </ol> === [[Linear Algebra/Introduction to Similarity|Similarity]]{{stage|100%|Jun 24, 2009}} === <ol type="I"> <li>[[Linear Algebra/Complex Vector Spaces|Complex Vector Spaces]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Factoring and Complex Numbers: A Review|Factoring and Complex Numbers: A Review]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Complex Representations|Complex Representations]]{{stage|100%|Jun 24, 2009}} <li>Similarity # [[Linear Algebra/Definition and Examples of Similarity|Definition and Examples]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Diagonalizability|Diagonalizability]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]{{stage|100%|Jun 24, 2009}} <li>[[Linear Algebra/Nilpotence|Nilpotence]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Self-Composition|Self-Composition]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Strings|Strings]]{{stage|100%|Jun 24, 2009}} <li>[[Linear Algebra/Jordan Form|Jordan Form]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Polynomials of Maps and Matrices|Polynomials of Maps and Matrices]]{{stage|100%|Jun 24, 2009}} #[[Linear Algebra/Jordan Canonical Form|Jordan Canonical Form]]{{stage|100%|Jun 24, 2009}} <li>[[Linear Algebra/Topic: Geometry of Eigenvalues|Topic: Geometry of Eigenvalues]]{{stage|50%|Jun 24, 2009}} <li>[[Linear Algebra/Topic: The Method of Powers|Topic: The Method of Powers]]{{stage|100%|Jun 24, 2009}} <li>[[Linear Algebra/Topic: Stable Populations|Topic: Stable Populations]]{{stage|100%|Jun 24, 2009}} <li>[[Linear Algebra/Topic: Linear Recurrences|Topic: Linear Recurrences]]{{stage|100%|Jun 24, 2009}} </ol> === Unitary Transformations === <ol type="I"> <li>[[Linear Algebra/Inner product spaces|Inner product spaces]]{{stage|75%}} <li>[[Linear Algebra/Unitary and Hermitian matrices|Unitary and Hermitian matrices]]{{stage|75%}} <li>[[Linear Algebra/Singular Value Decomposition|Singular Value Decomposition]]{{stage|75%}} <li>[[Linear Algebra/Spectral Theorem|Spectral Theorem]]{{stage|75%}} </ol> === [[Linear Algebra/Appendix|Appendix]] === *[[Linear Algebra/Propositions|Propositions]] *[[Linear Algebra/Quantifiers|Quantifiers]] *[[Linear Algebra/Techniques of Proof|Techniques of Proof]] *[[Linear Algebra/Sets, Functions, Relations|Sets, Functions, Relations]] ===封面=== <center> [[Image:Linalg_cover.png|alt=Several graphs distorted in a line|400px]] </center> ===符号说明=== <center> {| border="1px" cellpadding="5px" |- <TD><math> \mathbb{R} </math>, <math> \mathbb{R}^+ </math>, <math> \mathbb{R}^n </math> <TD>real numbers, reals greater than <math>0</math>, ordered <math>n</math>-tuples of reals |- | <math> \mathbb{N} </math> <TD> natural numbers: <math> \{0,1,2,\ldots\} </math> |- | <math> \mathbb{C} </math> | complex numbers |- | <math> \{\ldots\,\big|\,\ldots\} </math> | set of . . . such that . . . |- | <math> (a\,..\,b) </math>, <math> [a\,..\,b] </math> <TD>interval (open or closed) of reals between <math>a</math> and <math>b</math> |- | <math> \langle \ldots \rangle </math> | sequence; like a set but order matters |- | <math> V,W,U </math> | vector spaces |- | <math> \vec{v},\vec{w} </math> | vectors |- | <math>\vec{0}</math>, <math>\vec{0}_V</math> <TD>zero vector, zero vector of <math>V</math> |- | <math> B,D </math> | bases |- | <math> \mathcal{E}_n=\langle \vec{e}_1,\,\ldots,\,\vec{e}_n \rangle </math> <TD>standard basis for <math>\mathbb{R}^n</math> |- | <math> \vec{\beta},\vec{\delta} </math> | basis vectors |- | <math> {\rm Rep}_{B}(\vec{v}) </math> | matrix representing the vector |- | <math> \mathcal{P}_n </math> <TD>set of <math> n </math>-th degree polynomials |- | <math> \mathcal{M}_{n \! \times \! m} </math> <TD>set of <math> n \! \times \! m </math> matrices |- | <math> [S] </math> <TD>span of the set <math> S </math> |- | <math> M\oplus N </math> | direct sum of subspaces |- | <math> V\cong W </math> | isomorphic spaces |- | <math> h,g </math> | homomorphisms, linear maps |- | <math> H,G </math> | matrices |- | <math> t,s </math> | transformations; maps from a space to itself |- | <math> T,S </math> | square matrices |- | <math> {\rm Rep}_{B,D}(h) </math> <TD>matrix representing the map <math> h </math> |- | <math> h_{i,j} </math> <TD>matrix entry from row <math> i </math>, column <math> j </math> |- | <math> \left|T\right| </math> <TD>determinant of the matrix <math> T </math> 矩阵<math> T </math>的行列式 |- | <math> \mathcal{R}(h),\mathcal{N}(h) </math> <TD>rangespace and nullspace of the map <math> h </math> |- | <math> \mathcal{R}_\infty(h),\mathcal{N}_\infty(h) </math> | generalized rangespace and nullspace |} </center> ====Lower case Greek alphabet <br>小写希腊字母==== {{center| <math> \begin{array}{ll|ll|ll} \text{name} &\text{character} &\text{name} &\text{character} &\text{name} &\text{character} \\ \hline \text{alpha} & \alpha &\text{iota} & \iota &\text{rho} & \rho \\ \text{beta} & \beta &\text{kappa} & \kappa &\text{sigma} & \sigma \\ \text{gamma} & \gamma &\text{lambda} & \lambda &\text{tau} & \tau \\ \text{delta} & \delta &\text{mu} & \mu &\text{upsilon}& \upsilon \\ \text{epsilon} & \epsilon &\text{nu} & \nu &\text{phi} & \phi \\ \text{zeta} & \zeta &\text{xi} & \xi &\text{chi} & \chi \\ \text{eta} & \eta &\text{omicron}& o &\text{psi} & \psi \\ \text{theta} & \theta &\text{pi} & \pi &\text{omega} & \omega \end{array} </math>}} '''About the Cover.''' This is Cramer's Rule for the system <math>x_1+2x_2=6</math>, <math>3x_1+x_2=8</math>. The size of the first box is the determinant shown (the absolute value of the size is the area). The size of the second box is <math>x_1</math> times that, and equals the size of the final box. Hence, <math>x_1</math> is the final determinant divided by the first determinant. ===介绍=== This book helps students to master the material of a standard undergraduate linear algebra course. 这本书帮助学生掌握标准的本科线性代数课程的材料。 The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The audience is also standard: sophomores or juniors, usually with a background of at least one semester of calculus and perhaps with as much as three semesters. 材料是标准的，因为涵盖的主题是高斯约化，向量空间，线性映射，行列式，特征值和特征向量。听众也是标准的：大二或大三，通常至少有一个学期的微积分背景，也许有三个学期的时间。 The help that it gives to students comes from taking a developmental approach&mdash;this book's presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and extensive, careful, exercises. The developmental approach is what sets this book apart, so some expansion of the term is appropriate here. 它给学生的帮助来自于采取一种发展的方法这本书的介绍强调动机和自然性，由各种各样的例子和广泛的，仔细的，练习。发展的方法使这本书与众不同，所以在这里对这个术语进行一些扩展是合适的。 Courses in the beginning of most mathematics programs reward students less for understanding the theory and more for correctly applying formulas and algorithms. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underlay many mathematical investigations like elementary set and function facts, and a capacity for some independent reading and thinking. Linear algebra is an ideal spot to work on the transition between the two kinds of courses. It comes early in a program so that progress made here pays off later, but also comes late enough that students are often majors and minors. The material is coherent, accessible, and elegant. There are a variety of argument styles&mdash;proofs by contradiction, if and only if statements, and proofs by induction, for instance&mdash;and examples are plentiful. 大多数数学课程开始时的课程对学生的奖励较少，因为他们理解了理论，而更多的是因为他们正确地应用了公式和算法。以后的课程要求数学成熟：能够理解不同类型的论据，熟悉许多数学研究的主题，如基本集合和函数事实，以及独立阅读和思考的能力。线性代数是研究这两种课程之间过渡的理想场所。它在一个项目中出现得早，这样在这里取得的进步会得到回报，但也会来得太晚，以至于学生通常都是主修生和未成年学生。材料连贯，通俗易懂，优雅大方。用矛盾证明、当且仅当语句证明、归纳证明等多种论证方式，如实例和实例丰富。 So, the aim of this book's exposition is to help students develop from being successful at their present level, in classes where a majority of the members are interested mainly in applications in science or engineering, to being successful at the next level, that of serious students of the subject of mathematics itself. 因此，这本书的目的是帮助学生从目前的水平，在大多数成员主要对科学或工程应用感兴趣的课程中取得成功，发展到下一阶段的成功，即数学学科本身的严肃学生。 Helping students make this transition means taking the mathematics seriously, so all of the results in this book are proved. On the other hand, we cannot assume that students have already arrived, and so in contrast with more abstract texts, we give many examples and they are often quite detailed. 帮助学生完成这一转变意味着要认真对待数学，因此本书中的所有结果都得到了证明。另一方面，我们不能假设学生已经到了，因此与更抽象的文本相比，我们给出了许多例子，而且往往非常详细。 In the past, linear algebra texts commonly made this transition abruptly. They began with extensive computations of linear systems, matrix multiplications, and determinants. When the concepts&mdash;vector spaces and linear maps&mdash;finally appeared, and definitions and proofs started, often the change brought students to a stop. In this book, while we start with a computational topic, linear reduction, from the first we do more than compute. We do linear systems quickly but completely, including the proofs needed to justify what we are computing. Then, with the linear systems work as motivation and at a point where the study of linear combinations seems natural, the second chapter starts with the definition of a real vector space. This occurs by the end of the third week. 在过去，线性代数文本通常会突然进行这种转换。他们开始广泛计算线性系统，矩阵乘法和行列式。当概念向量空间和线性映射最终出现，定义和证明开始时，这种变化常常使学生停止。在这本书中，我们从一个计算主题开始，线性化简，从一开始我们做的不仅仅是计算。我们快速但完整地处理线性系统，包括证明我们正在计算的东西。然后，以线性系统为动力，在研究线性组合似乎很自然的地方，第二章从实向量空间的定义开始。这将在第三周结束时发生。 Another example of our emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomorphism, but with that of isomorphism. That's because this definition is easily motivated by the observation that some spaces are "just like" others. After that, the next section takes the reasonable step of defining homomorphism by isolating the operation-preservation idea. This approach loses mathematical slickness, but it is a good trade because it comes in return for a large gain in sensibility to students. 我们强调动机和自然性的另一个例子是，关于线性映射的第三章没有从同态的定义开始，而是从同构的定义开始。这是因为这个定义很容易被一些空间“和”其他空间“一样”的观察所激发。然后，下一节通过隔离操作保持的思想，采取合理的步骤来定义同态。这种方法失去了数学上的圆滑，但它是一种很好的交易，因为它可以让学生在情感上得到很大的提高。 One aim of a developmental approach is that students should feel throughout the presentation that they can see how the ideas arise, and perhaps picture themselves doing the same type of work. 发展性教学法的一个目的是让学生在整个演示过程中感觉到他们可以看到想法是如何产生的，也许还能想象自己在做同样类型的工作。 The clearest example of the developmental approach taken here&mdash;and the feature that most recommends this book&mdash;is the exercises. A student progresses most while doing the exercises, so they have been selected with great care. Each problem set ranges from simple checks to reasonably involved proofs. Since an instructor usually assigns about a dozen exercises after each lecture, each section ends with about twice that many, thereby providing a selection. There are even a few problems that are challenging puzzles taken from various journals, competitions, or problems collections. (These are marked with a "'''?'''" and as part of the fun, the original wording has been retained as much as possible.) In total, the exercises are aimed to both build an ability at, and help students experience the pleasure of, ''doing'' mathematics. 最清楚的例子，在这里采取的发展方法和特点，最推荐这本书是练习。学生在做练习时进步最大，所以他们是经过精心挑选的。每个习题集的范围从简单的检查到合理涉及的证明。由于教师通常在每堂课后布置十几个练习题，所以每节课结束时的练习数是原来的两倍，因此提供了一个选择题。甚至有一些问题是挑战性的难题从各种杂志，比赛，或问题收集。（这些标记有“？”总的来说，这些练习的目的是培养学生学习数学的能力，并帮助他们体验数学的乐趣。 ==== Applications and Computers <br>应用程序和计算机==== The point of view taken here, that linear algebra is about vector spaces and linear maps, is not taken to the complete exclusion of others. Applications and the role of the computer are important and vital aspects of the subject. Consequently, each of this book's chapters closes with a few application or computer-related topics. Some are: network flows, the speed and accuracy of computer linear reductions, Leontief Input/Output analysis, dimensional analysis, Markov chains, voting paradoxes, analytic projective geometry, and difference equations. 这里所采取的观点，即线性代数是关于向量空间和线性映射的，并不完全排除其他的观点。计算机的应用和作用是这门学科重要而重要的方面。因此，本书的每一章都以一些应用或计算机相关的主题结束。其中包括：网络流、计算机线性化简的速度和精度、Leontief输入/输出分析、量纲分析、马尔可夫链、投票悖论、解析射影几何和差分方程。 These topics are brief enough to be done in a day's class or to be given as independent projects for individuals or small groups. Most simply give the reader a taste of the subject, discuss how linear algebra comes in, point to some further reading, and give a few exercises. In short, these topics invite readers to see for themselves that linear algebra is a tool that a professional must have. 这些主题足够简短，可以在一天的课堂上完成，也可以作为个人或小组的独立项目。最简单的是让读者领略一下这个主题，讨论一下线性代数是如何产生的，指出一些进一步的阅读，并给出一些练习。简言之，这些主题邀请读者亲眼看到，线性代数是一个专业人士必须具备的工具。 ====For people reading this book on their own <br>对于自学这本书的人==== This book's emphasis on motivation and development make it a good choice for self-study. But, while a professional instructor can judge what pace and topics suit a class, if you are an independent student then perhaps you would find some advice helpful. 这本书对动机和发展的强调使它成为自学的好选择。但是，虽然专业的教师可以判断什么样的节奏和主题适合一节课，但如果你是一名独立学生，那么也许你会发现一些建议是有用的。 Here are two timetables for a semester. The first focuses on core material. 这是一个学期的两个时间表。第一个重点是核心材料。 <center> <TABLE border=1px> <TR> <TD style="border-bottom: 2px-black-solid">''week <br>星期'' <TD style="border-bottom: 2px-black-solid">''Monday <br>礼拜一'' <TD style="border-bottom: 2px-black-solid">''Wednesday <br>礼拜三'' <TD style="border-bottom: 2px-black-solid">''Friday <br>礼拜五'' <TR> <TD>1 <TD>One.I.1 <TD>One.I.1, 2 <TD>One.I.2, 3 <TR> <TD>2 <TD>One.I.3 <TD>One.II.1 <TD>One.II.2 <TR> <TD>3 <TD>One.III.1, 2 <TD>One.III.2 <TD>Two.I.1 <TR> <TD>4 <TD>Two.I.2 <TD>Two.II <TD>Two.III.1 <TR> <TD>5 <TD>Two.III.1, 2 <TD>Two.III.2 <TD>Exam <TR> <TD>6 <TD>Two.III.2, 3 <TD>Two.III.3 <TD>Three.I.1 <TR> <TD>7 <TD>Three.I.2 <TD>Three.II.1 <TD>Three.II.2 <TR> <TD>8 <TD>Three.II.2 <TD>Three.II.2 <TD>Three.III.1 <TR> <TD>9 <TD>Three.III.1 <TD>Three.III.2 <TD>Three.IV.1, 2 <TR> <TD>10 <TD>Three.IV.2, 3, 4 <TD>Three.IV.4 <TD>Exam <TR> <TD>11 <TD>Three.IV.4, Three.V.1 <TD>Three.V.1, 2 <TD>Four.I.1, 2 <TR> <TD>12 <TD>Four.I.3 <TD>Four.II <TD>Four.II <TR> <TD>13 <TD>Four.III.1 <TD>Five.I <TD>Five.II.1 <TR> <TD>14 <TD>Five.II.2 <TD>Five.II.3 <TD>Review </TABLE> </center> The second timetable is more ambitious (it supposes that you know One.II, the elements of vectors, usually covered in third semester calculus). 第二个时间表更具雄心（它假设你知道一、二，向量的元素，通常在第三学期微积分中讨论）。 <center> <TABLE border=1px> <TR> <TD>''week <br>星期'' <TD>''Monday <br>礼拜一'' <TD>''Wednesday <br>礼拜三'' <TD>''Friday <br>礼拜五'' <TR> <TD>1 <TD>One.I.1 <TD>One.I.2 <TD>One.I.3 <TR> <TD>2 <TD>One.I.3 <TD>One.III.1, 2 <TD>One.III.2 <TR> <TD>3 <TD>Two.I.1 <TD>Two.I.2 <TD>Two.II <TR> <TD>4 <TD>Two.III.1 <TD>Two.III.2 <TD>Two.III.3 <TR> <TD>5 <TD>Two.III.4 <TD>Three.I.1 <TD>Exam <TR> <TD>6 <TD>Three.I.2 <TD>Three.II.1 <TD>Three.II.2 <TR> <TD>7 <TD>Three.III.1 <TD>Three.III.2 <TD>Three.IV.1, 2 <TR> <TD>8 <TD>Three.IV.2 <TD>Three.IV.3 <TD>Three.IV.4 <TR> <TD>9 <TD>Three.V.1 <TD>Three.V.2 <TD>Three.VI.1 <TR> <TD>10 <TD>Three.VI.2 <TD>Four.I.1 <TD>Exam <TR> <TD>11 <TD>Four.I.2 <TD>Four.I.3 <TD>Four.I.4 <TR> <TD>12 <TD>Four.II <TD>Four.II, Four.III.1 <TD>Four.III.2, 3 <TR> <TD>13 <TD>Five.II.1, 2 <TD>Five.II.3 <TD>Five.III.1 <TR> <TD>14 <TD>Five.III.2 <TD>Five.IV.1, 2 <TD>Five.IV.2 </TABLE> </center> See the table of contents for the titles of these subsections. 这些小节的标题见目录。 To help you make time trade-offs, in the table of contents I have marked subsections as optional if some instructors will pass over them in favor of spending more time elsewhere. You might also try picking one or two topics that appeal to you from the end of each chapter. You'll get more from these if you have access to computer software that can do the big calculations. 为了帮助您进行时间权衡，在目录中，我将子部分标记为可选的，如果有些讲师会跳过它们，而将更多的时间花在其他地方。你也可以试着从每一章的结尾选一两个吸引你的话题。如果你能使用计算机软件进行大计算，你会从中得到更多。 The most important advice is: do many exercises. The recommended exercises are labeled throughout. (The answers are available.) You should be aware, however, that few inexperienced people can write correct proofs. Try to find a knowledgeable person to work with you on this. 最重要的建议是：多做练习。推荐的练习贯穿始终。（答案是有的）但是你应该知道，没有经验的人很少能写出正确的证明。试着找一个有见识的人和你一起工作。 Finally, if I may, a caution for all students, independent or not: I cannot overemphasize how much the statement that I sometimes hear, "I understand the material, but it's only that I have trouble with the problems" reveals a lack of understanding of what we are up to. Being able to do things with the ideas is their point. The quotes below express this sentiment admirably. They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general, and of linear algebra in particular (I took the liberty of formatting them as poems). 最后，如果可以的话，我要提醒所有的学生，不管他们是否独立：我不能过分强调我有时听到的“我理解材料，但只是我对问题有困难”这句话多少暴露了我们对我们正在做的事情缺乏了解。能够用这些想法做事是他们的重点。下面的引文很好地表达了这种观点。它们陈述了我认为是数学和科学的美和力量的关键，尤其是线性代数（我冒昧地将它们格式化为诗歌）。 {{quote |''I know of no better tactic''<br> ''&nbsp;than the illustration of exciting principles''<br> ''by well-chosen particulars.''<br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''--Stephen Jay Gould'' 我知道没有比这更好的策略了<br> 而不是那些令人兴奋的原则<br> 通过精心挑选的细节。<br> --史蒂芬·杰伊·古尔德 }} {{quote |''If you really wish to learn''<br> ''&nbsp;then you must mount the machine''<br> ''&nbsp;and become acquainted with its tricks''<br> ''by actual trial.''<br> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''--Wilbur Wright'' 如果你真的想学<br> 那你就得装上机器<br> 熟悉它的技巧<br> 通过实际试验。<br> --威尔伯·赖特 }} Jim Hefferon<br> Mathematics, Saint Michael's College<br> Colchester, Vermont USA 05439<br> <tt>http://joshua.smcvt.edu</tt><br> 2006-May-20 吉姆·赫弗伦 圣米歇尔学院数学系 美国佛蒙特州科尔切斯特市，邮编：05439 2006年5月20日 ''Author's Note.'' Inventing a good exercise, one that enlightens as well as tests, is a creative act, and hard work. 作者的笔记。创造一个好的练习，一个启发和测试，是一个创造性的行为，努力工作。 The inventor deserves recognition. But for some reason texts have traditionally not given attributions for questions. I have changed that here where I was sure of the source. I would greatly appreciate hearing from anyone who can help me to correctly attribute others of the questions. 这位发明家值得肯定。但由于某些原因，文本传统上没有给出问题的归属。我已经改变了这里我确定来源的地方。如果有人能帮助我正确回答其他问题，我将不胜感激。 ==第一章 Linear Systems/线性系统== ===第一节=== Systems of linear equations are common in science and mathematics. These two examples from high school science {{harv|O'Nan|1990}} give a sense of how they arise.<br> 线性方程组在科学和数学中很常见。这两个来自《高中科学》（奥南1990）的例子让我们了解了它们是如何产生的。 {{anchor|physics problem}}The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances.<br> 第一个例子来自物理学。假设我们得到三个物体，一个质量已知为2公斤，被要求找出未知质量。进一步假设，使用仪表棒进行试验可以产生这两种平衡。 <center> {|cellpadding=20px |[[Image:Linalg_balance_1.png|x130px]] |[[Image:Linalg_balance_2.png|x130px]] |} </center> Since the sum of magnitudes of the torques of the clockwise forces equal those of the counter clockwise forces (the torque of an object rotating about a fixed origin is the cross product of the force on it and its position vector relative to the origin; gravitational acceleration is uniform we can divide both sides by it). The two balances give this system of two equations.<br> 由于顺时针力的力矩大小之和等于逆时针力的大小（绕固定原点旋转的物体的力矩是其上的力与其相对于原点的位置矢量的叉积；重力加速度是均匀的，我们可以用它来划分两边）。这两个天平给出了这个由两个方程组成的系统。 :<math> \begin{cases} \begin{array}{rl} 40h+15c &= 100 \\ 25c &= 50+50h \end{array} \end{cases} </math> <quiz display=simple points=1/1> {Can you finish the solution?<br>你能解决这个问题吗？ |type="{}" coef="2"} c = { 4_2 } kg h = { 1_2 } kg </quiz> {{anchor|chemistry problem}}The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene <math>\hbox{C}_7\hbox{H}_8</math> and nitric acid <math>\hbox{H}\hbox{N}\hbox{O}_3</math> to produce trinitrotoluene <math>\hbox{C}_7\hbox{H}_5\hbox{O}_6\hbox{N}_3</math> along with the byproduct water (conditions have to be controlled very well, indeed&mdash; trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction 线性系统的第二个例子来自化学。我们可以在受控制的情况下混合甲苯（C₇H₈）和硝酸（HNO₃）的生产条件三硝基甲苯（C₇H₅O₆N₃）及其副产水（条件必须控制得很好，实际上，三硝基甲苯被称为TNT）。这些成分应该按多大比例混合？每种元素在反应前存在的原子数 :<math> x\,{\rm C}_7{\rm H}_8\ +\ y\,{\rm H}{\rm N}{\rm O}_3 \quad\longrightarrow\quad z\,{\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ w\,{\rm H}_2{\rm O} </math> must equal the number present afterward. Applying that principle to the elements C, H, N, and O in turn gives this system. 一定与反应后存在的原子数相等。根据这个反应中的C、H、N和O元素的守恒。 :<math>\begin{cases}\begin{array}{rl} 7x &= 7z \\ 8x +1y &= 5z+2w \\ 1y &= 3z \\ 3y &= 6z+1w \end{array}\end{cases}</math> <quiz display=simple points=1/1> {Can you balance the equation? 你能配平这个方程吗？ |type="{}" coef="4"} { 1_1 }<math>{\rm C}_7{\rm H}_8\ +\ </math>{ 3_1 }<math>{\rm H}{\rm N}{\rm O}_3\quad\longrightarrow\quad</math>{ 1_1 }<math>{\rm C}_7{\rm H}_5{\rm O}_6{\rm N}_3\ +\ </math>{ 3_1 }<math>{\rm H}_2{\rm O}</math> </quiz> To finish each of these examples requires solving a system of equations. In each, the equations involve only the first power of the variables. This chapter shows how to solve any such system. 要完成这些例子中的每一个都需要解一个方程组。在每一个方程中，方程只涉及变量的一次方。本章介绍如何解决任何此类系统。 <noinclude> =====References===== *{{citation|first1=Micheal|last1=O'Nan|title=Linear Algebra|publisher=Harcourt College Pub|edition=3rd|year=1990}}. ====§ 1.1 Gauss' Method 高斯消元法 ==== {{TextBox|1= ;Definition 1.1{{anchor|linear equation}}: A '''linear equation''' in variables <math>x_1,x_2,\ldots,x_n</math> has the form 未知数<math>x_1,x_2,\ldots,x_n</math>的一个线性方程组，形如 :<math>a_1x_1+a_2x_2+a_3x_3+\cdots+a_nx_n=d</math> where the numbers <math>a_1,\dots,a_n\in\R</math> are the equation's '''coefficients''' and <math>d\in\R</math> is the '''constant'''. An <math>n</math>-tuple <math>(s_1,s_2,\dots,s_n)\in\R^n </math> is a '''solution''' of, or '''satisfies''', that equation if substituting the numbers <math>s_1,\dots,s_n</math> for the variables gives a true statement: <math>a_1s_1+a_2s_2+\ldots+a_ns_n=d</math>. 其中<math>a_1,\dots,a_n\in\R</math>是方程的系数，<math>d\in\R</math>是一个常数项。 当一个<math>n</math>元数组<math>(s_1,s_2,\dots,s_n)\in\R^n </math>满足：<math>a_1s_1+a_2s_2+\ldots+a_ns_n=d</math>成立 那么我们称这个数组为为方程的一个解。 A '''system of linear equations''' 一个'''线性方程组''' :<math> \begin{array}{*{4}{rc}r} a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &+ &a_{1,n}x_n &= &d_1 \\ a_{2,1}x_1 &+ &a_{2,2}x_2 &+ &\cdots &+ &a_{2,n}x_n &= &d_2 \\ & & & & & & &\vdots \\ a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &+ &a_{m,n}x_n &= &d_m \end{array} </math> has the solution <math>(s_1,s_2,\ldots,s_n)</math> if that <math>n</math>-tuple is a solution of all of the equations in the system. 当有一个<math>n</math>元数组<math>(s_1,s_2,\dots,s_n)\in\R^n </math>，是方程组中每一个方程的解，那么我们称 这个<math>n</math>元数组为方程组的解。 }} {{TextBox|1= ;Example 1.2: The ordered pair <math>(-1,5)</math> is a solution of this system. :<math> \begin{array}{*{2}{rc}r} 4x_1 &+ &2x_2 &= &6 \\ -x_1 &+ &x_2 &= &6 \end{array} </math> In contrast, <math> (5,-1) </math> is not a solution. }} Finding the set of all solutions is '''solving'''{{anchor|system of linear equations!solving}} the system. No guesswork or good fortune is needed to solve a linear system. There is an algorithm that always works. The next example introduces that algorithm, called '''Gauss' method'''{{anchor|Gauss' method}}. It transforms the system, step by step, into one with a form that is easily solved. {{TextBox|1= ;Example 1.3: To solve this system :<math> \begin{array}{*{3}{rc}r} & & & &3x_3 &= &9 \\ x_1 &+ &5x_2 &- &2x_3 &= &2 \\ \frac{1}{3}x_1 &+ &2x_2 & & &= &3 \end{array} </math> we repeatedly transform it until it is in a form that is easy to solve. :<math>\begin{array}{rcl} \quad &\xrightarrow[]{ \text{swap row 1 with row 3} } &\begin{array}{*{3}{rc}r} \frac{1}{3}x_1 &+ &2x_2 & & &= &3 \\ x_1 &+ &5x_2 &- &2x_3 &= &2 \\ & & & &3x_3 &= &9 \end{array} \\ &\xrightarrow[]{ \text{multiply row 1 by 3} } &\begin{array}{*{3}{rc}r} x_1 &+ &6x_2 & & &= &9 \\ x_1 &+ &5x_2 &- &2x_3 &= &2 \\ & & & &3x_3 &= &9 \end{array} \\ &\xrightarrow[]{ \text{add }-1\text{ times row 1 to row 2} } &\begin{array}{*{3}{rc}r} x_1 &+ &6x_2 & & &= &9 \\ & &-x_2 &- &2x_3 &= &-7 \\ & & & &3x_3 &= &9 \end{array} \end{array} </math> The third step is the only nontrivial one. We've mentally multiplied both sides of the first row by <math>-1</math>, mentally added that to the old second row, and written the result in as the new second row. Now we can find the value of each variable. The bottom equation shows that <math>x_3=3</math>. Substituting <math>3</math> for <math>x_3</math> in the middle equation shows that <math>x_2=1</math>. Substituting those two into the top equation gives that <math>x_1=3</math> and so the system has a unique solution: the solution set is <math>\{\,(3,1,3)\,\}</math>. }} Most of this subsection and the next one consists of examples of solving linear systems by Gauss' method. We will use it throughout this book. It is fast and easy. But, before we get to those examples, we will first show that this method is also safe in that it never loses solutions or picks up extraneous solutions. {{TextBox|1= ;Theorem 1.4 (Gauss' method) {{anchor|th:GaussMethod}}:<!--\label{th:GaussMethod}--> If a linear system is changed to another by one of these operations <ol type=1 start=1> <li> an equation is swapped with another <li> an equation has both sides multiplied by a nonzero constant <li> an equation is replaced by the sum of itself and a multiple of another </ol> then the two systems have the same set of solutions. }} Each of those three operations has a restriction. Multiplying a row by <math>0</math> is not allowed because obviously that can change the solution set of the system. Similarly, adding a multiple of a row to itself is not allowed because adding <math>-1</math> times the row to itself has the effect of multiplying the row by <math>0</math>. Finally, swapping a row with itself is disallowed to make some results in the fourth chapter easier to state and remember (and besides, self-swapping doesn't accomplish anything). {{TextBox|1= ;Proof: We will cover the equation swap operation here and save the other two cases for [[#ex:ProveGaussMethod|Problem 14]]<!--\ref{ex:ProveGaussMethod}-->. Consider this swap of row <math>i</math> with row <math>j</math>. :<math> \begin{array}{*{4}{rc}r} a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &&a_{1,n}x_n &= &d_1 \\ & & & & & & &\vdots \\ a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &&a_{i,n}x_n &= &d_i \\ & & & & & & &\vdots \\ a_{j,1}x_1 &+ &a_{j,2}x_2 &+ &\cdots &&a_{j,n}x_n &= &d_j \\ & & & & & & &\vdots \\ a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &&a_{m,n}x_n &= &d_m \end{array} \xrightarrow[]{} \begin{array}{*{4}{rc}r} a_{1,1}x_1 &+ &a_{1,2}x_2 &+ &\cdots &&a_{1,n}x_n &= &d_1 \\ & & & & & & &\vdots \\ a_{j,1}x_1 &+ &a_{j,2}x_2 &+ &\cdots &&a_{j,n}x_n &= &d_j \\ & & & & & & &\vdots \\ a_{i,1}x_1 &+ &a_{i,2}x_2 &+ &\cdots &&a_{i,n}x_n &= &d_i \\ & & & & & & &\vdots \\ a_{m,1}x_1 &+ &a_{m,2}x_2 &+ &\cdots &&a_{m,n}x_n &= &d_m \end{array} </math> The <math>n</math>-tuple <math>(s_1,\ldots,s_n)</math> satisfies the system before the swap if and only if substituting the values, the <math>s</math>'s, for the variables, the <math>x</math>'s, gives true statements: <math>a_{1,1}s_1+a_{1,2}s_2+\cdots+a_{1,n}s_n=d_1</math> and ... <math>a_{i,1}s_1+a_{i,2}s_2+\cdots+a_{i,n}s_n=d_i</math> and ... <math>a_{j,1}s_1+a_{j,2}s_2+\cdots+a_{j,n}s_n=d_j</math> and ... <math>a_{m,1}s_1+a_{m,2}s_2+\cdots+a_{m,n}s_n=d_m</math>. In a requirement consisting of statements and-ed together we can rearrange the order of the statements, so that this requirement is met if and only if <math>a_{1,1}s_1+a_{1,2}s_2+\cdots+a_{1,n}s_n=d_1</math> and ... <math>a_{j,1}s_1+a_{j,2}s_2+\cdots+a_{j,n}s_n=d_j</math> and ... <math>a_{i,1}s_1+a_{i,2}s_2+\cdots+a_{i,n}s_n=d_i</math> and ... <math>a_{m,1}s_1+a_{m,2}s_2+\cdots+a_{m,n}s_n=d_m</math>. This is exactly the requirement that <math>(s_1,\ldots,s_n)</math> solves the system after the row swap. }} {{TextBox|1= ;Definition 1.5:{{anchor|elementary operations}} The three operations from [[#th:GaussMethod|Theorem 1.4]]<!--\ref{th:GaussMethod}--> are the '''elementary reduction operations''', or '''row operations''', or '''Gaussian operations'''. They are '''swapping''', '''multiplying by a scalar''' or '''rescaling''', and '''pivoting'''. }} When writing out the calculations, we will abbreviate "row <math>i</math>" by "<math>\rho_i</math>". For instance, we will denote a pivot operation by <math>k\rho_i+\rho_j</math>, with the row that is changed written second. We will also, to save writing, often list pivot steps together when they use the same <math>\rho_i</math>. {{TextBox|1= ;Example 1.6: A typical use of Gauss' method is to solve this system. :<math> \begin{array}{*{3}{rc}r} x &+ &y & & &= &0 \\ 2x &- &y &+ &3z &= &3 \\ x &- &2y &- &z &= &3 \end{array} </math> The first transformation of the system involves using the first row to eliminate the <math>x</math> in the second row and the <math>x</math> in the third. To get rid of the second row's <math>2x</math>, we multiply the entire first row by <math>-2</math>, add that to the second row, and write the result in as the new second row. To get rid of the third row's <math>x</math>, we multiply the first row by <math>-1</math>, add that to the third row, and write the result in as the new third row. :<math>\begin{array}{rcl} &\xrightarrow[-\rho_1+\rho_3]{-2\rho_1+\rho_2} &\begin{array}{*{3}{rc}r} x &+ &y & & &= &0 \\ & &-3y&+ &3z &= &3 \\ & &-3y&- &z &= &3 \end{array} \end{array} </math> (Note that the two <math>\rho_1</math> steps <math>-2\rho_1+\rho_2</math> and <math>-\rho_1+\rho_3</math> are written as one operation.) In this second system, the last two equations involve only two unknowns. To finish we transform the second system into a third system, where the last equation involves only one unknown. This transformation uses the second row to eliminate <math>y</math> from the third row. :<math>\begin{array}{rcl} &\xrightarrow[]{-\rho_2+\rho_3} &\begin{array}{*{3}{rc}r} x &+ &y & & &= &0 \\ & &-3y&+ &3z &= &3 \\ & & & &-4z&= &0 \end{array} \end{array} </math> Now we are set up for the solution. The third row shows that <math>z=0</math>. {{anchor|back substitution}}Substitute that back into the second row to get <math>y=-1</math>, and then substitute back into the first row to get <math> x=1 </math>. }} {{TextBox|1= ;Example 1.7{{anchor|statics problem}}: For the Physics problem from the start of this chapter, Gauss' method gives this. :<math>\begin{array}{rcl} \begin{array}{*{2}{rc}r} 40h &+ &15c &= &100 \\ -50h &+ &25c &= &50 \end{array} &\xrightarrow[]{5/4\rho_1 +\rho_2} &\begin{array}{*{2}{rc}r} 40h &+ &15c &= &100 \\ & &(175/4)c &= &175 \end{array} \end{array} </math> So <math> c=4 </math>, and back-substitution gives that <math> h=1 </math>. (The Chemistry problem is solved later.) }} {{TextBox|1= ;Example 1.8: The reduction :<math>\begin{array}{rcl} \begin{array}{*{3}{rc}r} x &+ &y &+ &z &= &9 \\ 2x &+ &4y &- &3z &= &1 \\ 3x &+ &6y &- &5z &= &0 \end{array} &\xrightarrow[-3\rho_1 +\rho_3]{-2\rho_1 +\rho_2} &\begin{array}{*{3}{rc}r} x &+ &y &+ &z &= &9 \\ & &2y &- &5z &= &-17\\ & &3y &- &8z&= &-27 \end{array} \\ &\xrightarrow[]{-(3/2)\rho_2+\rho_3} &\begin{array}{*{3}{rc}r} x &+ &y &+ &z &= &9 \\ & &2y &- &5z &= &-17\\ & & & &-(1/2)z &= &-(3/2) \end{array} \end{array} </math> shows that <math> z=3 </math>, <math> y=-1 </math>, and <math> x=7 </math>. }} As these examples illustrate, Gauss' method uses the elementary reduction operations to set up back-substitution. {{TextBox|1= ;Definition 1.9{{anchor|echelon form}}: In each row, the first variable with a nonzero coefficient is the row's '''leading variable'''. A system is in '''echelon form''' if each leading variable is to the right of the leading variable in the row above it (except for the leading variable in the first row). }} {{TextBox|1= ;Example 1.10: The only operation needed in the examples above is pivoting. Here is a linear system that requires the operation of swapping equations. After the first pivot :<math>\begin{array}{rcl} \begin{array}{*{4}{rc}r} x &- &y & & & & &= &0 \\ 2x &- &2y &+ &z &+ &2w &= &4 \\ & &y & & &+ &w &= &0 \\ & & & &2z &+ &w &= &5 \end{array} &\xrightarrow[]{-2\rho_1 +\rho_2} &\begin{array}{*{4}{rc}r} x &- &y & & & & &= &0 \\ & & & &z &+ &2w &= &4 \\ & &y & & &+ &w &= &0 \\ & & & &2z &+ &w &= &5 \end{array} \end{array} </math> the second equation has no leading <math>y</math>. To get one, we look lower down in the system for a row that has a leading <math>y</math> and swap it in. :<math>\begin{array}{rcl} &\xrightarrow[]{\rho_2 \leftrightarrow\rho_3} &\begin{array}{*{4}{rc}r} x &- &y & & & & &= &0 \\ & &y & & &+ &w &= &0 \\ & & & &z &+ &2w &= &4 \\ & & & &2z &+ &w &= &5 \end{array} \end{array} </math> (Had there been more than one row below the second with a leading <math>y</math> then we could have swapped in any one.) The rest of Gauss' method goes as before. :<math>\begin{array}{rcl} &\xrightarrow[]{-2\rho_3 +\rho_4} &\begin{array}{*{4}{rc}r} x &- &y & & & & &= &0 \\ & &y & & &+ &w &= &0 \\ & & & &z &+ &2w &= &4 \\ & & & & & &-3w&= &-3 \end{array} \end{array} </math> Back-substitution gives <math> w=1 </math>, <math> z=2 </math>, <math> y=-1 </math>, and <math> x=-1 </math>. }} Strictly speaking, the operation of rescaling rows is not needed to solve linear systems. We have included it because we will use it later in this chapter as part of a variation on Gauss' method, the Gauss-Jordan method. All of the systems seen so far have the same number of equations as unknowns. All of them have a solution, and for all of them there is only one solution. We finish this subsection by seeing for contrast some other things that can happen. {{TextBox|1= ;Example 1.11{{anchor|ex:MoreEqsThanUnks}}: <!--\label{ex:MoreEqsThanUnks}--> Linear systems need not have the same number of equations as unknowns. This system :<math> \begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ 2x &+ &y &= &-3 \\ 2x &+ &2y &= &-2 \end{array} </math> has more equations than variables. Gauss' method helps us understand this system also, since this :<math>\begin{array}{rcl} &\xrightarrow[-2\rho_1 +\rho_3]{-2\rho_1 +\rho_2} &\begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ & &-5y &= &-5 \\ & &-4y &= &-4 \end{array} \end{array} </math> shows that one of the equations is redundant. Echelon form :<math>\begin{array}{rcl} &\xrightarrow[]{-(4/5)\rho_2 +\rho_3} &\begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ & &-5y &= &-5 \\ & &0 &= &0 \end{array} \end{array} </math> gives <math> y=1 </math> and <math> x=-2 </math>. The "<math> 0=0 </math>" is derived from the redundancy. }} That example's system has more equations than variables. Gauss' method is also useful on systems with more variables than equations. Many examples are in the next subsection. Another way that linear systems can differ from the examples shown earlier is that some linear systems do not have a unique solution. This can happen in two ways. The first is that it can fail to have any solution at all. {{TextBox|1= ;Example 1.12{{anchor|ex:MoreEqsThanUnksInconsis}}: <!--\label{ex:MoreEqsThanUnksInconsis}--> Contrast the system in the last example with this one. :<math>\begin{array}{rcl} \begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ 2x &+ &y &= &-3 \\ 2x &+ &2y &= &0 \end{array} &\xrightarrow[-2\rho_1 +\rho_3]{-2\rho_1 +\rho_2} &\begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ & &-5y &= &-5 \\ & &-4y &= &-2 \end{array} \end{array} </math> Here the system is inconsistent: no pair of numbers satisfies all of the equations simultaneously. Echelon form makes this inconsistency obvious. :<math>\begin{array}{rcl} &\xrightarrow[]{-(4/5)\rho_2 +\rho_3} &\begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ & &-5y &= &-5 \\ & &0 &= &2 \end{array} \end{array} </math> The solution set is empty. }} {{TextBox|1= ;Example 1.13: The prior system has more equations than unknowns, but that is not what causes the inconsistency&mdash; [[#ex:MoreEqsThanUnks|Example 1.11]]<!--\ref{ex:MoreEqsThanUnks}--> has more equations than unknowns and yet is consistent. Nor is having more equations than unknowns sufficient for inconsistency, as is illustrated by this inconsistent system with the same number of equations as unknowns. :<math>\begin{array}{rcl} \begin{array}{*{2}{rc}r} x &+ &2y &= &8 \\ 2x &+ &4y &= &8 \end{array} &\xrightarrow[]{-2\rho_1 + \rho_2} &\begin{array}{*{2}{rc}r} x &+ &2y &= &8 \\ & &0 &= &-8 \end{array} \end{array} </math> }} The other way that a linear system can fail to have a unique solution is to have many solutions. {{TextBox|1= ;Example 1.14: In this system :<math> \begin{array}{*{2}{rc}r} x &+ &y &= &4 \\ 2x &+ &2y &= &8 \end{array} </math> any pair of numbers satisfying the first equation automatically satisfies the second. The solution set <math> \{ (x,y)\,\big|\, x+y=4 \} </math> is infinite; some of its members are <math>(0,4)</math>, <math>(-1,5)</math>, and <math>(2.5,1.5)</math>. The result of applying Gauss' method here contrasts with the prior example because we do not get a contradictory equation. :<math>\begin{array}{rcl} &\xrightarrow[]{-2\rho_1 + \rho_2} &\begin{array}{*{2}{rc}r} x &+ &y &= &4 \\ & &0 &= &0 \end{array} \end{array} </math> }} Don't be fooled by the "<math> 0=0 </math>" equation in that example. It is not the signal that a system has many solutions. {{TextBox|1= ;Example 1.15{{anchor|ex:NoZerosInfManySols}}: <!--\label{ex:NoZerosInfManySols}--> The absence of a "<math> 0=0 </math>" does not keep a system from having many different solutions. This system is in echelon form :<math> \begin{array}{*{3}{rc}r} x &+ &y &+ &z &= &0 \\ & &y &+ &z &= &0 \end{array} </math> has no "<math>0=0</math>", and yet has infinitely many solutions. (For instance, each of these is a solution: <math>(0,1,-1)</math>, <math>(0,1/2,-1/2)</math>, <math>(0,0,0)</math>, and <math>(0,-\pi,\pi)</math>. There are infinitely many solutions because any triple whose first component is <math>0</math> and whose second component is the negative of the third is a solution.) Nor does the presence of a "<math> 0=0 </math>" mean that the system must have many solutions. [[#ex:MoreEqsThanUnks|Example 1.11]]<!--\ref{ex:MoreEqsThanUnks}--> shows that. So does this system, which does not have many solutions&mdash; in fact it has none&mdash; despite that when it is brought to echelon form it has a "<math>0=0</math>" row. :<math>\begin{array}{rcl} \begin{array}{*{3}{rc}r} 2x & & &- &2z &= &6 \\ & &y &+ &z &= &1 \\ 2x &+ &y &- &z &= &7 \\ & &3y &+ &3z &= &0 \end{array} &\xrightarrow[]{-\rho_1 +\rho_3} &\begin{array}{*{3}{rc}r} 2x & & &- &2z &= &6 \\ & &y &+ &z &= &1 \\ & &y &+ &z &= &1 \\ & &3y &+ &3z &= &0 \end{array} \\ &\xrightarrow[-3\rho_2 +\rho_4]{-\rho_2 +\rho_3} &\begin{array}{*{3}{rc}r} 2x & & &- &2z &= &6 \\ & &y &+ &z &= &1 \\ & & & &0 &= &0 \\ & & & &0 &= &-3 \end{array} \end{array} </math> }} We will finish this subsection with a summary of what we've seen so far about Gauss' method. Gauss' method uses the three row operations to set a system up for back substitution. If any step shows a contradictory equation then we can stop with the conclusion that the system has no solutions. If we reach echelon form without a contradictory equation, and each variable is a leading variable in its row, then the system has a unique solution and we find it by back substitution. Finally, if we reach echelon form without a contradictory equation, and there is not a unique solution (at least one variable is not a leading variable) then the system has many solutions. The next subsection deals with the third case&mdash; we will see how to describe the solution set of a system with many solutions. == Exercises == {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 1: Use Gauss' method to find the unique solution for each system. <ol type=1 start=1> <li> <math>\begin{array}{*{2}{rc}r} 2x &+ &3y &= &13 \\ x &- &y &= &-1 \end{array}</math> <br><p><br><li> <math>\begin{array}{*{3}{rc}r} x & & &- &z &= &0 \\ 3x &+ &y & & &= &1 \\ -x &+ &y &+ &z &= &4 \end{array}</math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 2: Use Gauss' method to solve each system or conclude "many solutions" or "no solutions". <ol type=1 start=1> <li> <math> \begin{array}{*{2}{rc}r} 2x &+ &2y &= &5 \\ x &- &4y &= &0 \end{array} </math> <br><p><br> <li> <math> \begin{array}{*{2}{rc}r} -x &+ &y &= &1 \\ x &+ &y &= &2 \end{array} </math> <br><p><br> <li> <math> \begin{array}{*{3}{rc}r} x &- &3y &+ &z &= &1 \\ x &+ &y &+ &2z &= &14 \end{array} </math> <br><p><br> <li> <math> \begin{array}{*{2}{rc}r} -x &- &y &= &1 \\ -3x &- &3y &= &2 \end{array} </math> <br><p><br> <li> <math> \begin{array}{*{3}{rc}r} & &4y &+ &z &= &20 \\ 2x &- &2y &+ &z &= &0 \\ x & & &+ &z &= &5 \\ x &+ &y &- &z &= &10 \end{array} </math> <br><p><br> <li> <math> \begin{array}{*{4}{rc}r} 2x & & &+ &z &+ &w &= &5 \\ & &y & & &- &w &= &-1 \\ 3x & & &- &z &- &w &= &0 \\ 4x &+ &y &+ &2z &+ &w &= &9 \end{array} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 3: There are methods for solving linear systems other than Gauss' method. One often taught in high school is to solve one of the equations for a variable, then substitute the resulting expression into other equations. That step is repeated until there is an equation with only one variable. From that, the first number in the solution is derived, and then back-substitution can be done. This method takes longer than Gauss' method, since it involves more arithmetic operations, and is also more likely to lead to errors. To illustrate how it can lead to wrong conclusions, we will use the system :<math> \begin{array}{*{2}{rc}r} x &+ &3y &= &1 \\ 2x &+ &y &= &-3 \\ 2x &+ &2y &= &0 \end{array} </math> from [[#ex:MoreEqsThanUnksInconsis|Example 1.12]]<!--\ref{ex:MoreEqsThanUnksInconsis}-->. <ol type=1 start=1> <li> Solve the first equation for <math>x</math> and substitute that expression into the second equation. Find the resulting <math>y</math>. <li> Again solve the first equation for <math>x</math>, but this time substitute that expression into the third equation. Find this <math>y</math>. </ol> What extra step must a user of this method take to avoid erroneously concluding a system has a solution? }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 4: For which values of <math> k </math> are there no solutions, many solutions, or a unique solution to this system? :<math> \begin{array}{*{2}{rc}r} x &- &y &= &1 \\ 3x &- &3y &= &k \end{array} </math> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 5: This system is not linear, in some sense, :<math> \begin{array}{*{3}{rc}r} 2\sin\alpha &- &\cos\beta &+ &3\tan\gamma &= &3 \\ 4\sin\alpha &+ &2\cos\beta &- &2\tan\gamma &= &10 \\ 6\sin\alpha &- &3\cos\beta &+ &\tan\gamma &= &9 \end{array} </math> and yet we can nonetheless apply Gauss' method. Do so. Does the system have a solution? }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 6: What conditions must the constants, the <math>b</math>'s, satisfy so that each of these systems has a solution? ''Hint.'' Apply Gauss' method and see what happens to the right side {{harv|Anton|1987}}. <ol type=1 start=1> <li> <math> \begin{array}{*{2}{rc}r} x &- &3y &= &b_1 \\ 3x &+ &y &= &b_2 \\ x &+ &7y &= &b_3 \\ 2x &+ &4y &= &b_4 \end{array} </math> <br><p><br> <li> <math> \begin{array}{*{3}{rc}r} x_1 &+ &2x_2 &+ &3x_3 &= &b_1 \\ 2x_1 &+ &5x_2 &+ &3x_3 &= &b_2 \\ x_1 & & &+ &8x_3 &= &b_3 \end{array} </math> </ol> }} {{TextBox|1= ;Problem 7: True or false: a system with more unknowns than equations has at least one solution. (As always, to say "true" you must prove it, while to say "false" you must produce a counterexample.) }} {{TextBox|1= ;Problem 8{{anchor|chemistry problem}}: Must any Chemistry problem like the one that starts this subsection&mdash; a balance the reaction problem&mdash; have infinitely many solutions? }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 9: Find the coefficients <math> a </math>, <math> b </math>, and <math> c </math> so that the graph of <math> f(x)=ax^2+bx+c </math> passes through the points <math> (1,2) </math>, <math> (-1,6) </math>, and <math> (2,3) </math>. }} {{TextBox|1= ;Problem 10: Gauss' method works by combining the equations in a system to make new equations. <ol type=1 start=1> <li> Can the equation <math> 3x-2y=5 </math> be derived, by a sequence of Gaussian reduction steps, from the equations in this system? :<math> \begin{array}{*{2}{rc}r} x &+ &y &= &1 \\ 4x &- &y &= &6 \end{array} </math> <li> Can the equation <math> 5x-3y=2 </math> be derived, by a sequence of Gaussian reduction steps, from the equations in this system? :<math> \begin{array}{*{2}{rc}r} 2x &+ &2y &= &5 \\ 3x &+ &y &= &4 \end{array} </math> <li> Can the equation <math> 6x-9y+5z=-2 </math> be derived, by a sequence of Gaussian reduction steps, from the equations in the system? :<math> \begin{array}{*{3}{rc}r} 2x &+ &y &- &z &= &4 \\ 6x &- &3y &+ &z &= &5 \end{array} </math> </ol> }} {{TextBox|1= ;Problem 11: Prove that, where <math> a,b,\ldots,e </math> are real numbers and <math> a\neq 0 </math>, if :<math> ax+by=c </math> has the same solution set as :<math> ax+dy=e </math> then they are the same equation. What if <math> a=0 </math>? }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 12: Show that if <math> ad-bc\neq 0 </math> then :<math> \begin{array}{*{2}{rc}r} ax &+ &by &= &j \\ cx &+ &dy &= &k \end{array} </math> has a unique solution. }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 13: In the system :<math> \begin{array}{*{2}{rc}r} ax &+ &by &= &c \\ dx &+ &ey &= &f \end{array} </math> each of the equations describes a line in the <math> xy </math>-plane. By geometrical reasoning, show that there are three possibilities: there is a unique solution, there is no solution, and there are infinitely many solutions. }} {{TextBox|1= ;Problem 14 {{anchor|ex:ProveGaussMethod}}: <!--\label{ex:ProveGaussMethod}--> Finish the proof of [[#th:GaussMethod|Theorem 1.4]]<!--\ref{th:GaussMethod}-->. }} {{TextBox|1= ;Problem 15: Is there a two-unknowns linear system whose solution set is all of <math> \mathbb{R}^2 </math>? }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 16: Are any of the operations used in Gauss' method redundant? That is, can any of the operations be synthesized from the others? }} {{TextBox|1= ;Problem 17: Prove that each operation of Gauss' method is reversible. That is, show that if two systems are related by a row operation <math>S_1\rightarrow S_2</math> then there is a row operation to go back <math>S_2\rightarrow S_1</math>. }} {{TextBox|1= ;? Problem 18: A box holding pennies, nickels and dimes contains thirteen coins with a total value of <math> 83 </math> cents. How many coins of each type are in the box? {{harv|Anton|1987}} }} {{TextBox|1= ;? Problem 19: Four positive integers are given. Select any three of the integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers 29, 23, 21, and 17 are obtained. One of the original integers is: <ol type=1 start=1> <li> 19 <li> 21 <li> 23 <li> 29 <li> 17 </ol> {{harv|Salkind|1975|loc=1955 problem 38}} }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;? Problem 20: Laugh at this: <math> \mbox{AHAHA}+\mbox{TEHE}=\mbox{TEHAW} </math>. It resulted from substituting a code letter for each digit of a simple example in addition, and it is required to identify the letters and prove the solution unique {{harv|Ransom|Gupta|1935}}. }} {{TextBox|1= ;? Problem 21: The Wohascum County Board of Commissioners, which has 20 members, recently had to elect a President. There were three candidates (<math>A</math>, <math>B</math>, and <math>C</math>); on each ballot the three candidates were to be listed in order of preference, with no abstentions. It was found that 11 members, a majority, preferred <math>A</math> over <math>B</math> (thus the other 9 preferred <math>B</math> over <math>A</math>). Similarly, it was found that 12 members preferred <math>C</math> over <math>A</math>. Given these results, it was suggested that <math>B</math> should withdraw, to enable a runoff election between <math>A</math> and <math>C</math>. However, <math>B</math> protested, and it was then found that 14 members preferred <math>B</math> over <math>C</math>! The Board has not yet recovered from the resulting confusion. Given that every possible order of <math>A</math>, <math>B</math>, <math>C</math> appeared on at least one ballot, how many members voted for <math>B</math> as their first choice {{harv|Gilbert|Krusemeyer|Larson|1993|loc=Problem number 2}}? }} {{TextBox|1= ;? Problem 22: "This system of <math>n</math> linear equations with <math>n</math> unknowns," said the Great Mathematician, "has a curious property." "Good heavens!" said the Poor Nut, "What is it?" "Note," said the Great Mathematician, "that the constants are in arithmetic progression." "It's all so clear when you explain it!" said the Poor Nut. "Do you mean like <math> 6x+9y=12 </math> and <math> 15x+18y=21 </math>?" "Quite so," said the Great Mathematician, pulling out his bassoon. "Indeed, even larger systems can be solved regardless of their progression. Can you find their solution?" "Good heavens!" cried the Poor Nut, "I am baffled." Are you? {{harv|Dudley|Lebow|Rothman|1963}} }} <noinclude> ====§ 1.2 Describing the Solution Set 解集的表示 ==== A linear system with a unique solution has a solution set with one element. A linear system with no solution has a solution set that is empty. In these cases the solution set is easy to describe. Solution sets are a challenge to describe only when they contain many elements. {{TextBox|1= ;Example 2.1: This system has many solutions because in echelon form :<math>\begin{array}{rcl} \begin{array}{*{3}{rc}r} 2x & & &+ &z &= &3 \\ x &- &y &- &z &= &1 \\ 3x &- &y & & &= &4 \end{array} &\xrightarrow[-\left(\frac32\right)\rho_1+\rho_3]{-\left(\frac12\right)\rho_1+\rho_2} &\begin{array}{*{3}{rc}r} 2x & & &+ &z &= &3 \\ & &-y &- &\left(\frac32\right)z &= &-\frac12 \\ & &-y &- &\left(\frac32\right)z &= &-\frac12 \end{array} \\[3em] &\xrightarrow[]{-\rho_2+\rho_3} &\begin{array}{*{3}{rc}r} 2x & & &+ &z &= &3 \\ & &-y &- &\left(\frac32\right)z &= &-\frac12 \\ & & & &0 &= &0 \end{array} \end{array} </math> not all of the variables are leading variables. The Gauss' method theorem showed that a triple satisfies the first system if and only if it satisfies the third. Thus, the solution set <math>\Big\{(x,y,z)\Big|2x+z=3\text{ and }x-y-z=1\text{ and }3x-y=4\Big\}</math> can also be described as <math>\left\{(x,y,z)\Big|2x+z=3\text{ and }-y-\frac{3z}{2}=-\frac12\right\}</math> . However, this second description is not much of an improvement. It has two equations instead of three, but it still involves some hard-to-understand interaction among the variables. To get a description that is free of any such interaction, we take the variable that does not lead any equation, <math>z</math> , and use it to describe the variables that do lead, <math>x</math> and <math>y</math> . The second equation gives <math>y=\frac12-\frac32z</math> and the first equation gives <math>x=\frac32-\frac12z</math> . Thus, the solution set can be described as <math>\left\{(x,y,z)=\left(\frac32-\frac12z,\frac12-\frac32z,z\right)\Big|z\in\R\right\}</math> . For instance, <math>\left(\frac12,-\frac52,2\right)</math> is a solution because taking <math>z=2</math> gives a first component of <math>\frac12</math> and a second component of <math>-\frac52</math> . The advantage of this description over the ones above is that the only variable appearing, <math>z</math> , is unrestricted — it can be any real number. }} {{TextBox|1= ;Definition 2.2{{anchor|free variable}}: The non-leading variables in an echelon-form linear system are '''free variables'''. }} In the echelon form system derived in the above example, <math>x</math> and <math>y</math> are leading variables and <math>z</math> is free. {{TextBox|1= ;Example 2.3{{anchor|ex:Parametrize2}}: <!--\label{ex:Parametrize2}--> A linear system can end with more than one variable free. This row reduction :<math>\begin{array}{rcl} \begin{array}{*{4}{rc}r} x &+ &y &+ &z &- &w &= &1 \\ & &y &- &z &+ &w &= &-1 \\ 3x & & &+ &6z &- &6w &= &6 \\ & &-y &+ &z &- &w &= &1 \end{array} &\xrightarrow[]{-3\rho_1 +\rho_3} &\begin{array}{*{4}{rc}r} x &+ &y &+ &z &- &w &= &1 \\ & &y &- &z &+ &w &= &-1 \\ & &-3y &+ &3z &- &3w &= &3 \\ & &-y &+ &z &- &w &= &1 \end{array} \\[3em] &\xrightarrow[\rho_2+\rho_4]{3\rho_2+\rho_3} &\begin{array}{*{4}{rc}r} x &+ &y &+ &z &- &w &= &1 \\ & &y &- &z &+ &w &= &-1 \\ & & & & & &0 &= &0 \\ & & & & & &0 &= &0 \end{array} \end{array} </math> ends with <math>x</math> and <math>y</math> leading, and with both <math>z</math> and <math>w</math> free. To get the description that we prefer we will start at the bottom. We first express <math>y</math> in terms of the free variables <math>z</math> and <math>w</math> with <math>y=-1+z-w</math> . Next, moving up to the top equation, substituting for <math>y</math> in the first equation <math>x+(-1+z-w)+z-w=1</math> and solving for <math>x</math> yields <math>x=2-2z+2w</math> . Thus, the solution set is <math>\Big\{(2-2z+2w,-1+z-w,z,w)\Big|z,w\in\R\Big\}</math> . We prefer this description because the only variables that appear, <math>z</math> and <math>w</math> , are unrestricted. This makes the job of deciding which four-tuples are system solutions into an easy one. For instance, taking <math>z=1</math> and <math>w=2</math> gives the solution <math>(4,-2,1,2)</math> . In contrast, <math>(3,-2,1,2)</math> is not a solution, since the first component of any solution must be <math>2</math> minus twice the third component plus twice the fourth. }} {{TextBox|1= ;Example 2.4{{anchor|ex:Parametrize1}}: <!--\label{ex:Parametrize1}--> After this reduction :<math>\begin{array}{rcl} \begin{array}{*{4}{rc}r} 2x &- &2y & & & & &= &0 \\ & & & &z &+ &3w &= &2 \\ 3x &- &3y & & & & &= &0 \\ x &- &y &+ &2z &+ &6w &= &4 \end{array} &\xrightarrow[-(\frac12)\rho_1+\rho_4]{-(\frac32)\rho_1+\rho_3} &\begin{array}{*{4}{rc}r} 2x &- &2y & & & & &= &0 \\ & & & &z &+ &3w &= &2 \\ & & & & & &0 &= &0 \\ & & & &2z &+ &6w &= &4 \end{array} \\[3em] &\xrightarrow[]{-2\rho_2+\rho_4} &\begin{array}{*{4}{rc}r} 2x &- &2y & & & & &= &0 \\ & & & &z &+ &3w &= &2 \\ & & & & & &0 &= &0 \\ & & & & & &0 &= &0 \end{array} \end{array} </math> <math>x,z</math> lead, <math>y,w</math> are free. The solution set is <math>\Big\{(y,y,2-3w,w)\Big|y,w\in\R\Big\}</math> . For instance, <math>(1,1,2,0)</math> satisfies the system — take <math>y=1</math> and <math>w=0</math> . The four-tuple <math>(1,0,5,4)</math> is not a solution since its first coordinate does not equal its second. }} {{anchor|parameter}} We refer to a variable used to describe a family of solutions as a '''parameter''' and we say that the set above is '''parametrized''' with <math>y</math> and <math>w</math> . (The terms "parameter" and "free variable" do not mean the same thing. Above, <math>y</math> and <math>w</math> are free because in the echelon form system they do not lead any row. They are parameters because they are used in the solution set description. We could have instead parametrized with <math>y</math> and <math>z</math> by rewriting the second equation as <math>w=\frac23-\frac13z</math> . In that case, the free variables are still <math>y</math> and <math>w</math> , but the parameters are <math>y</math> and <math>z</math> . Notice that we could not have parametrized with <math>x</math> and <math>y</math> , so there is sometimes a restriction on the choice of parameters. The terms "parameter" and "free" are related because, as we shall show later in this chapter, the solution set of a system can always be parametrized with the free variables. Consequently, we shall parametrize all of our descriptions in this way.) {{TextBox|1= ;Example 2.5: This is another system with infinitely many solutions. :<math>\begin{array}{rcl} \begin{array}{*{4}{rc}r} x &+ &2y & & & & &= &1 \\ 2x & & &+ &z & & &= &2 \\ 3x &+ &2y &+ &z &- &w &= &4 \end{array} &\xrightarrow[-3\rho_1 +\rho_3]{-2\rho_1+\rho_2} &\begin{array}{*{4}{rc}r} x &+ &2y & & & & &= &1 \\ & &-4y &+ &z & & &= &0 \\ & &-4y &+ &z &- &w &= &1 \end{array} \\[3em] &\xrightarrow[]{-\rho_2+\rho_3} &\begin{array}{*{4}{rc}r} x &+ &2y & & & & &= &1 \\ & &-4y &+ &z & & &= &0 \\ & & & & & &-w &= &1 \end{array} \end{array} </math> The leading variables are <math>x,y,w</math> . The variable <math>z</math> is free. (Notice here that, although there are infinitely many solutions, the value of one of the variables is fixed — <math>w=-1</math> .) Write <math>w</math> in terms of <math>z</math> with <math>w=-1+0z</math> . Then <math>y=\frac14z</math> . To express <math>x</math> in terms of <math>z</math> , substitute for <math>y</math> into the first equation to get <math>x=1-\frac12z</math>. The solution set is <math>\left\{\left(1-\frac12z,\frac14z,z,-1\right)\Bigg|z\in\R\right\}</math> . }} We finish this subsection by developing the notation for linear systems and their solution sets that we shall use in the rest of this book. {{TextBox|1= ;Definition 2.6{{anchor|matrix}}: An <math>m\times n</math> '''matrix''' is a rectangular array of numbers with <math>m</math> '''rows''' and <math>n</math> '''columns'''. Each number in the matrix is an '''entry'''. }} Matrices are usually named by upper case roman letters, e.g. <math>A</math> . Each entry is denoted by the corresponding lower-case letter, e.g. <math>a_{i,j}</math> is the number in row <math>i</math> and column <math>j</math> of the array. For instance, :<math>A=\begin{pmatrix}1&2.2&5\\3&4&-7\end{pmatrix}</math> has two rows and three columns, and so is a <math>2\times3</math> matrix. (Read that "two-by-three"; the number of rows is always stated first.) The entry in the second row and first column is <math>a_{2,1}=3</math> . Note that the order of the subscripts matters: <math>a_{1,2}\ne a_{2,1}</math> since <math>a_{1,2}=2.2</math> . (The parentheses around the array are a typographic device so that when two matrices are side by side we can tell where one ends and the other starts.) Matrices occur throughout this book. We shall use <math>\mathcal{M}_{n\times m}</math> to denote the collection of <math>n\times m</math> matrices. {{TextBox|1= ;Example 2.7: We can abbreviate this linear system :<math> \begin{array}{*{3}{rc}r} x &+ &2y & & &= &4 \\ & &y &- &z &= &0 \\ x & & &+ &2z&= &4 \end{array} </math> with this matrix. :<math> \left(\begin{array}{ccc|c} 1 &2 &0 &4 \\ 0 &1 &-1 &0 \\ 1 &0 &2 &4 \end{array}\right) </math> The vertical bar just reminds a reader of the difference between the coefficients on the systems's left hand side and the constants on the right. {{anchor|augmented matrix}}When a bar is used to divide a matrix into parts, we call it an '''augmented''' matrix. In this notation, Gauss' method goes this way. :<math> \left(\begin{array}{ccc|c} 1 &2 &0 &4 \\ 0 &1 &-1 &0 \\ 1 &0 &2 &4 \end{array}\right) \xrightarrow[]{-\rho_1 +\rho_3} \left(\begin{array}{ccc|c} 1 &2 &0 &4 \\ 0 &1 &-1 &0 \\ 0 &-2 &2 &0 \end{array}\right) \xrightarrow[]{2\rho_2 +\rho_3} \left(\begin{array}{ccc|c} 1 &2 &0 &4 \\ 0 &1 &-1 &0 \\ 0 &0 &0 &0 \end{array}\right) </math> The second row stands for <math>y-z=0</math> and the first row stands for <math>x+2y=4</math> so the solution set is <math>\Big\{(4-2z,z,z)\Big|z\in\R\Big\}</math> . One advantage of the new notation is that the clerical load of Gauss' method — the copying of variables, the writing of <math>+</math>'s and <math>=</math>'s, etc. — is lighter. }} We will also use the array notation to clarify the descriptions of solution sets. A description like <math>\{(2-2z+2w,-1+z-w,z,w)\big|z,w\in\R\}</math> from [[#ex:Parametrize2|Example 2.3]]<!--\ref{ex:Parametrize2}--> is hard to read. We will rewrite it to group all the constants together, all the coefficients of <math>z</math> together, and all the coefficients of <math>w</math> together. We will write them vertically, in one-column wide matrices. :<math> \left\{\begin{pmatrix}2\\-1\\0\\0\end{pmatrix}+\begin{pmatrix}-2\\1\\1\\0\end{pmatrix}z+\begin{pmatrix}2\\-1\\0\\1\end{pmatrix}w\Bigg|z,w\in\R\right\} </math> For instance, the top line says that <math>x=2-2z+2w</math> . The next section gives a geometric interpretation that will help us picture the solution sets when they are written in this way. {{TextBox|1= ;Definition 2.8{{anchor|vector}}: A '''vector''' (or '''column vector''') is a matrix with a single column. A matrix with a single row is a '''row vector'''. The entries of a vector are its '''components'''. }} Vectors are an exception to the convention of representing matrices with capital roman letters. We use lower-case roman or greek letters overlined with an arrow: <math>\vec a,\vec b</math> ... or <math>\vec{\alpha},\vec{\beta}</math> ... (boldface is also common: <math>\mathbf{a}</math> or <math> \boldsymbol{\alpha}</math>). For instance, this is a column vector with a third component of <math>7</math> . :<math>\vec v=\begin{pmatrix}1\\3\\7\end{pmatrix}</math> {{TextBox|1= ;Definition 2.9{{anchor|vector satisfies}}: The linear equation <math>a_1x_1+\cdots+a_nx_n=d</math> with unknowns <math> x_1,\ldots\,,x_n </math> is '''satisfied''' by :<math>\vec s=\begin{pmatrix}s_1\\ \vdots\\s_n\end{pmatrix}</math> if <math>a_1s_1+\cdots+a_ns_n=d</math> . A vector satisfies a linear system if it satisfies each equation in the system. }} The style of description of solution sets that we use involves adding the vectors, and also multiplying them by real numbers, such as the <math>z</math> and <math>w</math> . We need to define these operations. {{TextBox|1= ;Definition 2.10{{anchor|vector sum}}: The '''vector sum''' of <math>\vec u</math> and <math>\vec v</math> is this. :<math> \vec u+\vec v=\begin{pmatrix}u_1\\ \vdots\\u_n\end{pmatrix}+\begin{pmatrix}v_1\\ \vdots\\v_n\end{pmatrix}=\begin{pmatrix}u_1+v_1\\ \vdots\\u_n+v_n\end{pmatrix} </math> In general, two matrices with the same number of rows and the same number of columns add in this way, entry-by-entry. }} {{TextBox|1= ;Definition 2.11{{anchor|scalar multiplication}}: The '''scalar multiplication''' of the real number <math>r</math> and the vector <math>\vec v</math> is this. :<math>r\cdot\vec v=r\cdot\begin{pmatrix}v_1\\ \vdots\\v_n\end{pmatrix}=\begin{pmatrix}rv_1\\ \vdots\\rv_n\end{pmatrix}</math> In general, any matrix is multiplied by a real number in this entry-by-entry way. }} Scalar multiplication can be written in either order: <math>r\cdot\vec v</math> or <math>\vec v\cdot r</math> , or without the "<math>\cdot</math>" symbol: <math>r\vec v</math> . (Do not refer to scalar multiplication as "scalar product" because that name is used for a different operation.) {{TextBox|1= ;Example 2.12: :<math> \begin{pmatrix}2\\3\\1\end{pmatrix}+\begin{pmatrix}3\\-1\\4\end{pmatrix}=\begin{pmatrix}2+3\\3-1\\1+4\end{pmatrix}=\begin{pmatrix}5\\2\\5\end{pmatrix} \qquad 7\cdot\begin{pmatrix}1\\4\\-1\\-3\end{pmatrix}=\begin{pmatrix}7\\28\\-7\\-21\end{pmatrix} </math> }} Notice that the definitions of vector addition and scalar multiplication agree where they overlap, for instance, <math>\vec v+\vec v=2\vec v</math> . With the notation defined, we can now solve systems in the way that we will use throughout this book. {{TextBox|1= ;Example 2.13{{anchor|ex:ManyParamsInfManySolsSystem}}: <!--\label{ex:ManyParamsInfManySolsSystem}--> This system :<math> \begin{array}{*{5}{rc}r} 2x &+ &y & & &- &w & & &= &4 \\ & &y & & &+ &w &+ &u &= &4 \\ x & & &- &z &+ &2w & & &= &0 \end{array} </math> reduces in this way. :<math>\begin{array}{rcl} \left(\begin{array}{ccccc|c} 2 &1 &0 &-1 &0 &4 \\ 0 &1 &0 &1 &1 &4 \\ 1 &0 &-1 &2 &0 &0 \end{array}\right) &\xrightarrow[]{-\left(\frac12\right)\rho_1+\rho_3} &\left(\begin{array}{ccccc|c} 2 &1 &0 &-1 &0 &4 \\ 0 &1 &0 &1 &1 &4 \\ 0 &-\frac12 &-1 &\frac52 &0 &-2 \end{array}\right) \\[3em] &\xrightarrow[]{\left(\frac12\right)\rho_2+\rho_3} &\left(\begin{array}{ccccc|c} 2 &1 &0 &-1 &0 &4 \\ 0 &1 &0 &1 &1 &4 \\ 0 &0 &-1 &3 &\frac12 &0 \end{array}\right) \end{array} </math> The solution set is <math>\left\{(w+\frac12u,4-w-u,3w+\frac12u,w,u)\Bigg|w,u\in\R\right\}</math> . We write that in vector form. :<math> \left\{\begin{pmatrix} x \\ y \\ z \\ w \\ u \end{pmatrix}= \begin{pmatrix} 0 \\ 4 \\ 0 \\ 0 \\ 0 \end{pmatrix}+ \begin{pmatrix} 1 \\ -1 \\ 3 \\ 1 \\ 0 \end{pmatrix}w+ \begin{pmatrix} \frac12 \\ -1 \\ \frac12 \\ 0 \\ 1 \end{pmatrix}u \Bigg|w,u\in\R\right\} </math> Note again how well vector notation sets off the coefficients of each parameter. For instance, the third row of the vector form shows plainly that if <math>u</math> is held fixed then <math> z </math> increases three times as fast as <math>w</math> . That format also shows plainly that there are infinitely many solutions. For example, we can fix <math>u</math> as <math>0</math> , let <math>w</math> range over the real numbers, and consider the first component <math>x</math> . We get infinitely many first components and hence infinitely many solutions. Another thing shown plainly is that setting both <math>w,u</math> to 0 gives that this :<math>\begin{pmatrix}x\\y\\z\\w\\u\end{pmatrix}=\begin{pmatrix}0\\4\\0\\0\\0\end{pmatrix}</math> is a particular solution of the linear system. }} {{TextBox|1= ;Example 2.14: In the same way, this system :<math> \begin{array}{*{3}{rc}r} x &- &y &+ &z &= &1 \\ 3x & & &+ &z &= &3 \\ 5x &- &2y &+ &3z &= &5 \end{array} </math> reduces :<math> \left(\begin{array}{ccc|c} 1 &-1 &1 &1 \\ 3 &0 &1 &3 \\ 5 &-2 &3 &5 \end{array}\right) \xrightarrow[-5\rho_1+\rho_3]{-3\rho_1+\rho_2} \left(\begin{array}{ccc|c} 1 &-1 &1 &1 \\ 0 &3 &-2 &0 \\ 0 &3 &-2 &0 \end{array}\right) \xrightarrow[]{-\rho_2+\rho_3} \left(\begin{array}{ccc|c} 1 &-1 &1 &1 \\ 0 &3 &-2 &0 \\ 0 &0 &0 &0 \end{array}\right) </math> to a one-parameter solution set. :<math>\left\{\begin{pmatrix}1\\0\\0\end{pmatrix}+\begin{pmatrix}-\frac13\\ \frac23\\1\end{pmatrix}z\Bigg|z\in\R\right\}</math> }} Before the exercises, we pause to point out some things that we have yet to do. The first two subsections have been on the mechanics of Gauss' method. Except for one result, [[Linear Algebra/Gauss' Method#th:GaussMethod|Theorem 1.4]]<!--\ref{th:GaussMethod}-->&mdash; without which developing the method doesn't make sense since it says that the method gives the right answers&mdash; we have not stopped to consider any of the interesting questions that arise. For example, can we always describe solution sets as above, with a particular solution vector added to an unrestricted linear combination of some other vectors? The solution sets we described with unrestricted parameters were easily seen to have infinitely many solutions so an answer to this question could tell us something about the size of solution sets. An answer to that question could also help us picture the solution sets, in <math>\mathbb{R}^2</math>, or in <math>\mathbb{R}^3</math>, etc. Many questions arise from the observation that Gauss' method can be done in more than one way (for instance, when swapping rows, we may have a choice of which row to swap with). [[Linear Algebra/Gauss' Method#th:GaussMethod|Theorem 1.4]]<!--\ref{th:GaussMethod}--> says that we must get the same solution set no matter how we proceed, but if we do Gauss' method in two different ways must we get the same number of free variables both times, so that any two solution set descriptions have the same number of parameters? Must those be the same variables (e.g., is it impossible to solve a problem one way and get <math>y</math> and <math>w</math> free or solve it another way and get <math>y</math> and <math>z</math> free)? In the rest of this chapter we answer these questions. The answer to each is "yes". The first question is answered in the last subsection of this section. In the second section we give a geometric description of solution sets. In the final section of this chapter we tackle the last set of questions. Consequently, by the end of the first chapter we will not only have a solid grounding in the practice of Gauss' method, we will also have a solid grounding in the theory. We will be sure of what can and cannot happen in a reduction. == Exercises == {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 1: Find the indicated entry of the matrix, if it is defined. :<math> A=\begin{pmatrix} 1 &3 &1 \\ 2 &-1 &4 \end{pmatrix} </math> <ol type=1 start=1> <li> <math> a_{2,1} </math> <li> <math> a_{1,2} </math> <li> <math> a_{2,2} </math> <li> <math> a_{3,1} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 2: Give the size of each matrix. <ol type=1 start=1> <li> <math> \begin{pmatrix} 1 &0 &4 \\ 2 &1 &5 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 1 &1 \\ -1 &1 \\ 3 &-1 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 5 &10 \\ 10 &5 \end{pmatrix} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 3: Do the indicated vector operation, if it is defined. <ol type=1 start=1> <li> <math> \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix} +\begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix} </math> <li> <math> 5\begin{pmatrix} 4 \\ -1 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 1 \\ 5 \\ 1 \end{pmatrix} -\begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix} </math> <li> <math> 7\begin{pmatrix} 2 \\ 1 \end{pmatrix} +9\begin{pmatrix} 3 \\ 5 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 1 \\ 2 \end{pmatrix} +\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} </math> <li> <math> 6\begin{pmatrix} 3 \\ 1 \\ 1 \end{pmatrix} -4\begin{pmatrix} 2 \\ 0 \\ 3 \end{pmatrix} +2\begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 4: Solve each system using matrix notation. Express the solution using vectors. <ol type=1 start=1> <li> <math> \begin{array}{*{2}{rc}r} 3x &+ &6y &= &18 \\ x &+ &2y &= &6 \end{array} </math> <li> <math> \begin{array}{*{2}{rc}r} x &+ &y &= &1 \\ x &- &y &= &-1 \end{array} </math> <li> <math> \begin{array}{*{3}{rc}r} x_1 & & &+ &x_3 &= &4 \\ x_1 &- &x_2 &+ &2x_3 &= &5 \\ 4x_1 &- &x_2 &+ &5x_3 &= &17 \end{array} </math> <li> <math> \begin{array}{*{3}{rc}r} 2a &+ &b &- &c &= &2 \\ 2a & & &+ &c &= &3 \\ a &- &b & & &= &0 \end{array} </math> <li> <math> \begin{array}{*{4}{rc}r} x &+ &2y &- &z & & &= &3 \\ 2x &+ &y & & &+ &w &= &4 \\ x &- &y &+ &z &+ &w &= &1 \end{array} </math> <li> <math> \begin{array}{*{4}{rc}r} x & & &+ &z &+ &w &= &4 \\ 2x &+ &y & & &- &w &= &2 \\ 3x &+ &y &+ &z & & &= &7 \end{array} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 5{{anchor|exer:SlvMatNot}}: <!--\label{exer:SlvMatNot}--> Solve each system using matrix notation. Give each solution set in vector notation. <ol type=1 start=1> <li> <math> \begin{array}{*{3}{rc}r} 2x &+ &y &- &z &= &1 \\ 4x &- &y & & &= &3 \end{array} </math> <li> <math> \begin{array}{*{4}{rc}r} x & & &- &z & & &= &1 \\ & &y &+ &2z &- &w &= &3 \\ x &+ &2y &+ &3z &- &w &= &7 \end{array} </math> <li> <math> \begin{array}{*{4}{rc}r} x &- &y &+ &z & & &= &0 \\ & &y & & &+ &w &= &0 \\ 3x &- &2y &+ &3z &+ &w &= &0 \\ & &-y & & &- &w &= &0 \end{array} </math> <li> <math> \begin{array}{*{5}{rc}r} a &+ &2b &+ &3c &+ &d &- &e &= &1 \\ 3a &- &b &+ &c &+ &d &+ &e &= &3 \end{array} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 6: The vector is in the set. What value of the parameters produces that vector? <ol type=1 start=1> <li> <math>\begin{pmatrix} 5 \\ -5 \end{pmatrix}</math>, <math>\{\begin{pmatrix} 1 \\ -1 \end{pmatrix}k\,\big|\, k\in\mathbb{R}\}</math> <li> <math>\begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}</math>, <math>\{\begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}i +\begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}j\,\big|\, i,j\in\mathbb{R}\}</math> <li> <math>\begin{pmatrix} 0 \\ -4 \\ 2 \end{pmatrix}</math>, <math>\{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}m +\begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}n\,\big|\, m,n\in\mathbb{R}\}</math> </ol> }} {{TextBox|1= ;Problem 7: Decide if the vector is in the set. <ol type=1 start=1> <li> <math>\begin{pmatrix} 3 \\ -1 \end{pmatrix}</math>, <math>\{\begin{pmatrix} -6 \\ 2 \end{pmatrix}k\,\big|\, k\in\mathbb{R}\}</math> <li> <math>\begin{pmatrix} 5 \\ 4 \end{pmatrix}</math>, <math>\{\begin{pmatrix} 5 \\ -4 \end{pmatrix}j\,\big|\, j\in\mathbb{R}\}</math> <li> <math>\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}</math>, <math>\{\begin{pmatrix} 0 \\ 3 \\ -7 \end{pmatrix}+\begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}r\,\big|\, r\in\mathbb{R}\}</math> <li> <math>\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}</math>, <math>\{\begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}j +\begin{pmatrix} -3 \\ -1 \\ 1 \end{pmatrix}k\,\big|\, j,k\in\mathbb{R}\}</math> </ol> }} {{TextBox|1= ;Problem 8: Parametrize the solution set of this one-equation system. :<math> x_1+x_2+\cdots+x_n=0 </math> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 9: <ol type=1 start=1> <li> Apply Gauss' method to the left-hand side to solve :<math> \begin{array}{*{4}{rc}r} x &+ &2y & & &- &w &= &a \\ 2x & & &+ &z & & &= &b \\ x &+ &y & & &+ &2w &= &c \end{array} </math> for <math> x </math>, <math> y </math>, <math> z </math>, and <math> w </math>, in terms of the constants <math>a</math>, <math>b</math>, and <math>c</math>. Note that <math> w </math> will be a free variable. <li> Use your answer from the prior part to solve this. :<math> \begin{array}{*{4}{rc}r} x &+ &2y & & &- &w &= &3 \\ 2x & & &+ &z & & &= &1 \\ x &+ &y & & &+ &2w &= &-2 \end{array} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 10: Why is the comma needed in the notation "<math> a_{i,j} </math>" for matrix entries? }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 11: Give the <math> 4 \! \times \! 4 </math> matrix whose <math> i,j </math>-th entry is <ol type=1 start=1> <li> <math> i+j </math>; <li> <math> -1 </math> to the <math> i+j </math> power. </ol> }} {{TextBox|1= ;Problem 12{{anchor|transpose}}: For any matrix <math> A </math>, the '''transpose''' of <math> A </math>, written <math> {{A}^{\rm trans}} </math>, is the matrix whose columns are the rows of <math> A </math>. Find the transpose of each of these. <ol type=1 start=1> <li> <math> \begin{pmatrix} 1 &2 &3 \\ 4 &5 &6 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 2 &-3 \\ 1 &1 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 5 &10 \\ 10 &5 \end{pmatrix} </math> <li> <math> \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} </math> </ol> }} {{Linear Algebra/Book 2/Recommended}} {{TextBox|1= ;Problem 13: <ol type=1 start=1> <li> Describe all functions <math> f(x)=ax^2+bx+c </math> such that <math> f(1)=2 </math> and <math> f(-1)=6 </math>. <li> Describe all functions <math> f(x)=ax^2+bx+c </math> such that <math> f(1)=2 </math>. </ol> }} {{TextBox|1= ;Problem 14: Show that any set of five points from the plane <math> \mathbb{R}^2 </math> lie on a common conic section, that is, they all satisfy some equation of the form <math> ax^2+by^2+cxy+dx+ey+f=0 </math> where some of <math> a,\,\ldots\,,f </math> are nonzero. }} {{TextBox|1= ;Problem 15: Make up a four equations/four unknowns system having <ol type=1 start=1> <li> a one-parameter solution set; <li> a two-parameter solution set; <li> a three-parameter solution set. </ol> }} {{TextBox|1= ;? Problem 16: <ol type=1 start=1> <li> Solve the system of equations. :<math> \begin{array}{*{2}{rc}r} ax &+ &y &= &a^2 \\ x &+ &ay &= &1 \end{array} </math> For what values of <math>a</math> does the system fail to have solutions, and for what values of <math>a</math> are there infinitely many solutions? <li> Answer the above question for the system. :<math> \begin{array}{*{2}{rc}r} ax &+ &y &= &a^3 \\ x &+ &ay &= &1 \end{array} </math> </ol> ''([[#USSROlympiad174|USSR Olympiad #174]])'' }} {{TextBox|1= ;? Problem 17: In air a gold-surfaced sphere weighs <math> 7588 </math> grams. It is known that it may contain one or more of the metals aluminum, copper, silver, or lead. When weighed successively under standard conditions in water, benzene, alcohol, and glycerine its respective weights are <math> 6588 </math>, <math> 6688 </math>, <math> 6778 </math>, and <math> 6328 </math> grams. How much, if any, of the forenamed metals does it contain if the specific gravities of the designated substances are taken to be as follows? <center> <TABLE border=0px cellpadding=10px> <TR> <TD>Aluminum <TD> 2.7 <TD> <TD> <TD>Alcohol <TD>0.81 <TR> <TD>Copper <TD> 8.9 <TD> <TD> <TD>Benzene <TD>0.90 <TR> <TD>Gold <TD> 19.3 <TD> <TD> <TD>Glycerine <TD> 1.26 <TR> <TD> Lead <TD> 11.3 <TD> <TD> <TD>Water <TD>1.00 <TR> <TD>Silver <TD> 10.8 </TABLE> </center> {{harv|Duncan|Quelch|1952}} }} hd50g4ovsn7r1oj1g5gswx9wnu8l128 4 32161 181525 181512 2025-06-14T23:01:28Z Cewbot 42738 [[User:Cewbot/log/20170915/configuration|生成議題列表：36個議題]] 181525 wikitext text/x-wiki <!-- 本頁面由機器人自動更新。若要改進，請聯繫機器人操作者。 --> {| class="wikitable sortable mw-collapsible" style="float:left;" |- ! data-sort-type="number" style="font-weight: normal;" | <small>#</small> !! 💭 話題 !! <span title="發言數/發言人次 (實際上為計算簽名數)">💬</span> !! <span title="參與討論人數/發言人數">👥</span> !! 🙋 最新發言 !! data-sort-type="isoDate" | <span title="最後更新">🕒 <small>(UTC+8)</small></span> |- | style="text-align: right;" | 1 | [[:Wikibooks:互助客棧#再谈改名建议|再谈改名建议]] | style="text-align: right;background-color: #ffe;" | 31 | style="text-align: right;" | 18 | style="background-color: #bbb;" | [[User:Joyance-Tsui|Joyance-Tsui]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2024-12-09T13:27:00.000Z" | 2024-12-09 <span style="color: blue;">21:27</span> |- | style="text-align: right;" | 2 | [[:Wikibooks:互助客棧#Unblock-zh.org|Unblock-zh.org]] | style="text-align: right;background-color: #ffe;" | 10 | style="text-align: right;" | 3 | style="background-color: #bbb;" | [[User:Bluedeck|Bluedeck]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2024-06-12T18:43:00.000Z" | 2024-06-13 <span style="color: blue;">02:43</span> |- | style="text-align: right;" | 3 | [[:Wikibooks:互助客棧#请求删除过时的小工具|请求删除过时的小工具]] | style="text-align: right;" | 4 | style="text-align: right;" | 4 | [[User:WhitePhosphorus|WhitePhosphorus]] | data-sort-type="isoDate" data-sort-value="2025-06-08T17:59:00.000Z" | 2025-06-09 <span style="color: blue;">01:59</span> |- | style="text-align: right;" | 4 | [[:Wikibooks:互助客棧#AdvancedSiteNotices禁止外部品牌連結？|AdvancedSiteNotices禁止外部品牌連結？]] | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:Kitabc12345|Kitabc12345]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2024-09-20T18:06:00.000Z" | 2024-09-21 <span style="color: blue;">02:06</span> |- | style="text-align: right;" | 5 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#'Wikidata_item'_link_is_moving._Find_out_where...|&apos;Wikidata item&apos; link is moving. 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word-break: break-all;">MediaWiki message delivery</small>]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-02-22T08:28:00.000Z" | 2025-02-22 <span style="color: blue;">16:28</span> |- | style="text-align: right;" | 21 | [[:Wikibooks:互助客棧#台灣分會2025年2月對話時間|台灣分會2025年2月對話時間]] | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:MediaWiki message delivery|<small style="word-wrap: break-word; word-break: break-all;">MediaWiki message delivery</small>]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-02-24T14:08:00.000Z" | 2025-02-24 <span style="color: blue;">22:08</span> |- | style="text-align: right;" | 22 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#Universal_Code_of_Conduct_annual_review:_proposed_changes_are_available_for_comment|Universal Code of Conduct annual review: proposed changes are available for comment]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:Keegan (WMF)|Keegan (WMF)]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-03-07T18:50:00.000Z" | 2025-03-08 <span style="color: blue;">02:50</span> |- | style="text-align: right;" | 23 | [[:Wikibooks:互助客棧#本wiki即將切換至只讀模式|本wiki即將切換至只讀模式]] | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:MediaWiki message delivery|<small style="word-wrap: break-word; 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word-break: break-all;">MediaWiki message delivery</small>]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-04-26T01:16:00.000Z" | 2025-04-26 <span style="color: blue;">09:16</span> |- | style="text-align: right;" | 28 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#對《通用行為準則執行規範及其協調委員會章程》的修訂投票|對《通用行為準則執行規範及其協調委員會章程》的修訂投票]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:Keegan (WMF)|Keegan (WMF)]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-04-29T03:40:00.000Z" | 2025-04-29 <span style="color: blue;">11:40</span> |- | style="text-align: right;" | 29 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#We_will_be_enabling_the_new_Charts_extension_on_your_wiki_soon!|We will be enabling the new Charts extension on your wiki soon!]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:Sannita (WMF)|Sannita (WMF)]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-05-06T15:07:00.000Z" | 2025-05-06 <span style="color: blue;">23:07</span> |- | style="text-align: right;" | 30 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#徵求《通用行為準則》協調委員會（U4C）候選人|徵求《通用行為準則》協調委員會（U4C）候選人]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #bbb;" | [[User:Keegan (WMF)|Keegan (WMF)]] | style="background-color: #bbb;" data-sort-type="isoDate" data-sort-value="2025-05-15T22:06:00.000Z" | 2025-05-16 <span style="color: blue;">06:06</span> |- | style="text-align: right;" | 31 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#RfC_ongoing_regarding_Abstract_Wikipedia_(and_your_project)|RfC ongoing regarding Abstract Wikipedia (and your project)]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #ddd;" | [[User:Sannita (WMF)|Sannita (WMF)]] | style="background-color: #ddd;" data-sort-type="isoDate" data-sort-value="2025-05-22T15:26:00.000Z" | 2025-05-22 <span style="color: blue;">23:26</span> |- | style="text-align: right;" | 32 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#2025年維基媒體基金會理事會理事選舉公告及徵求提問|2025年維基媒體基金會理事會理事選舉公告及徵求提問]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #ddd;" | [[User:MediaWiki message delivery|<small style="word-wrap: break-word; word-break: break-all;">MediaWiki message delivery</small>]] | style="background-color: #ddd;" data-sort-type="isoDate" data-sort-value="2025-05-28T03:07:00.000Z" | 2025-05-28 <span style="color: blue;">11:07</span> |- | style="text-align: right;" | 33 | [[:Wikibooks:互助客棧#台灣分會2025年5月對話時間|台灣分會2025年5月對話時間]] | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #ddd;" | [[User:MediaWiki message delivery|<small style="word-wrap: break-word; word-break: break-all;">MediaWiki message delivery</small>]] | style="background-color: #ddd;" data-sort-type="isoDate" data-sort-value="2025-05-29T03:06:00.000Z" | 2025-05-29 <span style="color: blue;">11:06</span> |- | style="text-align: right;" | 34 | style="max-width: 24em" | <small>[[:Wikibooks:互助客棧#请求将道本语加入template:MAIN_PAGE_CHOICENESS|请求将道本语加入template:MAIN PAGE CHOICENESS]]</small> | style="text-align: right;background-color: #fcc;" | 1 | style="text-align: right;background-color: #fcc;" | 1 | style="background-color: #ddd;" | [[User:Kijetesantakalu pulaso|<small style="word-wrap: break-word; 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