中国科学院指定考研参考书 線性代數 李炯生查建国编著 中国科学技术大学出版社 1989
中 国 科 学 院 指 定 考 研 参 考 书 線 性 代 數 李炯生 查建国 编著 中 国 科 学 技 术 大 学 出 版 社 1 9 8 9
内容提要 本书是作者在中国科学技术大学数学系多年教学的基础上编写而 成的.它由多项式、行列式、矩阵、线性空问、线性变换、Jordan标准形 Euclid空间、酉空间和双线性函数等九章组成.在内容的叙述上，力图 做到矩阵方法与几何方法并重每章都配有丰富的典型例题和充足的习 题可供读者选用.的 附录中收录龚昇数授编著的《线性代数五讲》，从现代数学，尤其是 模论的观点来重新审视与认识线性代数，讨论了向量空间、线性变换，着 重研究了主理想督环上的模及其分解，并以此来重新理解向量空间在线 性算子作用下的分解，可以使读者从高一个层次上来认识线性代数。 本书适合作为综合性大学理科数学专业的教材，也可以作为各类大 专院校师生的教学参考书，以及关心线性代数与矩阵论的科技工作者和 学爱好者的白学读物或参考书。心 图书在版编目(CP)数据 线性代数李炯生，查建国编著.一合肥：中国科学 技术大学出版社，1989(2005.9重印，2010.10重排) (中国科学院指定考研参考书) Is8N978-731206116-9 线0类2查.m线推代数一高等学校 教材V0151.2 中国版本图韩馆CP数据核字(203)第078690号 郑重声明 《线性代数》版权归中国科学技术大学出版社所有， 本重排本仅限用于个人学习和XTX排版技术交 流，请勿用于任何商业行为，因私自散布造成的法律及 相关问题，重排者一律不予负责！本书已有第二版发 行，全国各大书店均应有售，请支持购买正版！ 版权归原出版社所有侵权必究
内 容 提 要 本书是作者在中国科学技术大学数学系多年教学的基础上编写而 成的．它由多项式、行列式、矩阵、线性空间、线性变换、Jordan 标准形、 Euclid 空间、酉空间和双线性函数等九章组成．在内容的叙述上，力图 做到矩阵方法与几何方法并重．每章都配有丰富的典型例题和充足的习 题可供读者选用． 附录中收录龚昇教授编著的《线性代数五讲》，从现代数学，尤其是 模论的观点来重新审视与认识线性代数，讨论了向量空间、线性变换，着 重研究了主理想整环上的模及其分解，并以此来重新理解向量空间在线 性算子作用下的分解，可以使读者从高一个层次上来认识线性代数． 本书适合作为综合性大学理科数学专业的教材，也可以作为各类大 专院校师生的教学参考书，以及关心线性代数与矩阵论的科技工作者和 数学爱好者的自学读物或参考书． 图书在版编目 (CIP) 数据 线性代数 / 李炯生，查建国编著．—合肥：中国科学 技术大学出版社，1989（2005.9 重印，2010.10 重排） （中国科学院指定考研参考书） ISBN 978-7-312-00110-9 I. 线. II. ① 李. ② 查. III. 线性代数—高等学校 —教材 IV. O 151.2 中国版本图书馆 CIP 数据核字（2003）第 078690 号 郑 重 声 明 《线性代数》版权归中国科学技术大学出版社所有， 本重排本仅限用于个人学习和 XƎLATEX 排版技术交 流，请勿用于任何商业行为，因私自散布造成的法律及 相关问题，重排者一律不予负责！本书已有第二版发 行，全国各大书店均应有售，请支持、购买正版！ 版权归原出版社所有 侵权必究
序言 本书初稿完成于1983年.当时中国科学技术大学数学系领导冯克 勤教授委托编著者编写一本供数学系用的线性代数讲义.接受这项任 务后，我们专程到北京，拜访了中国科学院系统科学研究所万哲先研究 员、中国科学院数学研完所许以超研突员、北京大学数学系聂灵沼教授 和中国科学院研究生院曾骨成教拎，清款他们对数学系线性代数教学 的设想.他们郁热情地给子指导，从而为编写讲父挺供了坐实的基础 1984年春天，讲义便开始在数学系83级使用，并作为数学系线性代数 教材一直使用到现在.1985年，讲义曾获得中国科学技术大学首次领 发的优秀教村一等势.此后，在使用过程中对讲义风作了进一步的修改」 出版前编著者又作了全面的加工和充实. 线性代数研究的是线性空间以及线性空间的线性变换在线性空 间取定一组基下，线性变换便和矩阵建立了A吧对应关系这样，线性 变换就和矩阵紧密联系起来.于是，研究线性空间以及线性空间关于线 性变换的分解即构成了线性代数的几何理论，而研究矩阵在各种关系下 的分类问题则是线性代数的代数理论.本书编写的一个着眼点是，着力 于建立线性代数的这两大理论之间的联系，并从这种联系去闸述线性代 数的理论。 当然，线性代数内容非常年富，本书尽灭能地接照1980年教有部 领发的综合性大学理科数学专业高等代数教学大纲进行选择，并在体系 安排与叙述方式人量吸收中国科学技术大学数学系长期从事线性代 数教学的老师与同事们，持别是曾肯成款、许以超研究员的教学经验 在处理行列天理论时采用了曾肯成教授1965年首先在中国科学技术 大学数学系使用的将阶行列式视为数域F上的n雏向量空间的规范 反对称重线性函数的讲清在处理线性方程组理论时，利用了矩阵在 相抵下的标准形理论；在处理Jordan标准形时，先考虑了线性空间关 于线性变换的分解，然后再用纯矩阵方法处理了Jordan标准形.同时 也着重于阐述已故著名数学家华罗庚教授的独具特色的矩阵方法
序 言 本书初稿完成于 1983 年．当时中国科学技术大学数学系领导冯克 勤教授委托编著者编写一本供数学系用的线性代数讲义．接受这项任 务后，我们专程到北京，拜访了中国科学院系统科学研究所万哲先研究 员、中国科学院数学研究所许以超研究员、北京大学数学系聂灵沼教授 和中国科学院研究生院曾肯成教授，请教他们对数学系线性代数教学 的设想．他们都热情地给予指导，从而为编写讲义提供了坚实的基础． 1984 年春天，讲义便开始在数学系 83 级使用，并作为数学系线性代数 教材一直使用到现在．1985 年，讲义曾获得中国科学技术大学首次颁 发的优秀教材一等奖．此后，在使用过程中对讲义又作了进一步的修改． 出版前编著者又作了全面的加工和充实． 线性代数研究的是线性空间以及线性空间的线性变换．在线性空 间取定一组基下，线性变换便和矩阵建立了一一对应关系．这样，线性 变换就和矩阵紧密联系起来．于是，研究线性空间以及线性空间关于线 性变换的分解即构成了线性代数的几何理论，而研究矩阵在各种关系下 的分类问题则是线性代数的代数理论．本书编写的一个着眼点是，着力 于建立线性代数的这两大理论之间的联系，并从这种联系去阐述线性代 数的理论． 当然，线性代数内容非常丰富，本书尽可能地按照 1980 年教育部 颁发的综合性大学理科数学专业高等代数教学大纲进行选择，并在体系 安排与叙述方式上尽量吸收中国科学技术大学数学系长期从事线性代 数教学的老师与同事们，特别是曾肯成教授、许以超研究员的教学经验． 在处理行列式理论时，采用了曾肯成教授 1965 年首先在中国科学技术 大学数学系使用的将 n 阶行列式视为数域 F 上的 n 维向量空间的规范 反对称 n 重线性函数的讲法；在处理线性方程组理论时，利用了矩阵在 相抵下的标准形理论；在处理 Jordan 标准形时，先考虑了线性空间关 于线性变换的分解，然后再用纯矩阵方法处理了 Jordan 标准形．同时 也着重于阐述已故著名数学家华罗庚教授的独具特色的矩阵方法．
I. 序言州 为了便于读者掌握解题技巧，提高分析问题、解决问题的能力，本书 几乎每一章都专门用一节讲述各种典型例题.这部分内容是为习题课 安排的.每一节后面都附有大量习题，教师与读者可以根据具体情况选 择使用.这些习题除了在众多的线性代数、矩降论教材以及习题集中选 取之外，有一些是取自我国历届研究生试题，还有一些是自己编撰的.在 教学过程中，有些同学对线性代数的其些课题产生了兴趣，进行了一些 研究.有些结果即成为本书的越，这里应该提到的有中国科学技术大 学数学系81级同学陈贵忠、黄瑜、案昌柱，82级同学陈秀雄等 冯克勤款授对本书的编写自始至终都给子了热情的关心与帮助 在编写过程中，得到万哲先许以急、聂灵沼、曾肯成等研究员和教授的 热心指导，编者谨致衷心感谢.中国科学技术大学数学系陆洪文教授 杜锡录、李尚志副款授曾经使用本书的前身一《线性代数讲义》 作为教材他们对讲义的修改提出许多有益的意见.中国科学技术大学 数学系讲师层善辣、徐俊明协助编者仔细地审核了原稿，安徽大学数学 系夏悬宠同志、中国科学技术大学86级硕士研究生黄道德审技了习题， 在此并致谢 由于编著者水平所限，错误与缺点在所难免.热忧欢迎同行们和广 李炯生查建国 九八从年元钥于合肥
⋅ II ⋅ 序 言 ¹ 为了便于读者掌握解题技巧，提高分析问题、解决问题的能力，本书 几乎每一章都专门用一节讲述各种典型例题．这部分内容是为习题课 安排的．每一节后面都附有大量习题，教师与读者可以根据具体情况选 择使用．这些习题除了在众多的线性代数、矩阵论教材以及习题集中选 取之外，有一些是取自我国历届研究生试题，还有一些是自己编撰的．在 教学过程中，有些同学对线性代数的某些课题产生了兴趣，进行了一些 研究．有些结果即成为本书的习题．这里应该提到的有中国科学技术大 学数学系 81 级同学陈贵忠、黄瑜、窦昌柱，82 级同学陈秀雄等． 冯克勤教授对本书的编写自始至终都给予了热情的关心与帮助． 在编写过程中，得到万哲先、许以超、聂灵沼、曾肯成等研究员和教授的 热心指导，编者谨致衷心感谢．中国科学技术大学数学系陆洪文教授， 杜锡录、李尚志副教授曾经使用本书的前身——《线性代数讲义》—— 作为教材，他们对讲义的修改提出许多有益的意见．中国科学技术大学 数学系讲师屈善坤、徐俊明协助编者仔细地审核了原稿，安徽大学数学 系夏恩虎同志、中国科学技术大学 86 级硕士研究生黄道德审核了习题， 在此一并致谢． 由于编著者水平所限，错误与缺点在所难免．热忱欢迎同行们和广 大读者批评指正． 李炯生 查建国 一九八八年元月于合肥
目 录 序言.1 第一章多项式 。1 s1.1 整数环与数域 s1.2 一元多项式环 4 51.3 整除性与最大公因式 6 51.4 唯一析因定理 17 51.5 实系数与复系数多项式 51.6整系数与有理系教多项式 517多元多项式环······ 51.8 对称多项式 第二章行列式 。 。 52.1数域F上的n雏向童空间， 52.2n阶行列式的定义与性质 52.3 Laplace展开定理····· 54 52.4 Cramer法则 64 52.5行列式的计算 第三章矩阵··· 82 53.1矩阵的代数运算 82 532 Binet-Cauchy公式 92 533可逆矩阵· 534矩阵的秩与相抓 535 116 536线姓方程组 125 53.矩陈的广义 133 第四章线性空间： 140 54.1线性空间的定义 140 54.2 线性相关 145 54.3 基与坐标 152 54.4 基查换与坐标变换 157 160
目 录 序 言 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ I 第一章 多项式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 §1.1 整数环与数域 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 §1.2 一元多项式环 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 4 §1.3 整除性与最大公因式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 6 §1.4 唯一析因定理 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 17 §1.5 实系数与复系数多项式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 20 §1.6 整系数与有理系数多项式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 23 §1.7 多元多项式环 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 27 §1.8 对称多项式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 30 第二章 行列式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 38 §2.1 数域 F上的 n 维向量空间 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 38 §2.2 n 阶行列式的定义与性质 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 43 §2.3 Laplace 展开定理 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 54 §2.4 Cramer 法则 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 64 §2.5 行列式的计算 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 68 第三章 矩阵 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 82 §3.1 矩阵的代数运算 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 82 §3.2 Binet-Cauchy 公式 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 92 §3.3 可逆矩阵 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 99 §3.4 矩阵的秩与相抵 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 107 §3.5 一些例子 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 116 §3.6 线性方程组 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 125 §3.7 矩阵的广义逆 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 133 第四章 线性空间 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 140 §4.1 线性空间的定义 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 140 §4.2 线性相关 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 145 §4.3 基与坐标 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 152 §4.4 基变换与坐标变换 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 157 §4.5 同构 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 160