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线性代数主讲:方聪娜理学院应用数学教研室联系方式:电话:13306012625课程简介:线性代数是讨论代数学中线性关系经典理论的课程,它具有较强的抽象性和逻辑性,是高等学校经管类各专业的一门重要的基础理论课。对线性方程组的讨论,在理论上和历史上都是线性代数这门学科的起点。由于线性问题广泛存在于科学技术的各个领域,而某些非线性问题在一定条件下,可以转化为线性问题,因此本课程所介绍的思想和方法广泛地应用于各个学科。本学期课程包括以下内容:行列式、矩阵、向量、线性方程组、矩阵的特征值与特征向量。课程特点:1、是一门基础课程,为后续课程做准备.2、定义、定理、推论繁多,必须理解记忆和区别.3、具有较强的抽象性和逻辑性.参考书目:《线性代数》(第三版)赵树嫄编中国人民大学出版社《实用线性代数》郑昌明编中国人民大学出版社行列式是线性代数的一个重要组成部分.它是研究矩阵、线性方程组、特征多项式的重要工具.本章从二、三阶行列式出发,介绍了n阶行列式的概念、性质、计算方法.第1章行列式用消元法解二元线性方程组.,22221211212111bxaxabxaxa12:122a,2212221212211abxaaxaa:212a,1222221212112abxaaxaa,得两式相减消去2x一、二阶行列式的引入;212221121122211baabxaaaa)(,得类似地,消去1x,211211221122211abbaxaaaa)(时,当021122211aaaa方程组的解为,211222112122211aaaabaabx)(3.211222112112112aaaaabbax由方程组的四个系数确定.由四个数排成二行二列(横排称行、竖排称列)的数表)4(22211211aaaa定义)5(42221121121122211aaaaaaaa行列式,并记作)所确定的二阶称为数表(表达式即.2112221122211211aaaaaaaaD11a12a22a12a主对角线副对角线对角线法则2211aa.2112aa二阶行列式的计算若记,22211211aaaaD.,22221211212111bxaxabxaxa对于二元线性方程组系数行列式.,22221211212111bxaxabxaxa,22211211aaaaD.,22221211212111bxaxabxaxa,2221211ababD.,22221211212111bxaxabxaxa,22211211aaaaD.,22221211212111bxaxabxaxa,2221211ababD.,22221211212111bxaxabxaxa.2211112babaD则二元线性方程组的解为,2221121122212111aaaaababDDx注意分母都为原方程组的系数行列式..2221121122111122aaaababaDDx例1.12,12232121xxxx求解二元线性方程组解1223D)4(3,07112121D,14121232D,21DDx11,2714DDx22.3721二、三阶行列式定义333231232221131211)5(339aaaaaaaaa列的数表行个数排成设有记,312213332112322311322113312312332211)6(aaaaaaaaaaaaaaaaaa333231232221131211aaaaaaaaa(6)式称为数表(5)所确定的三阶行列式.323122211211aaaaaa.312213332112322311aaaaaaaaa(1)沙路法三阶行列式的计算322113312312332211aaaaaaaaaD333231232221131211aaaaaaaaaD.列标行标333231232221131211aaaaaaaaaD333231232221131211aaaaaaaaa332211aaa.322311aaa(2)对角线法则注意红线上三元素的乘积冠以正号,蓝线上三元素的乘积冠以负号.说明1对角线法则只适用于二阶与三阶行列式.322113aaa312312aaa312213aaa332112aaa如果三元线性方程组;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa的系数行列式333231232221131211aaaaaaaaaD,0利用三阶行列式求解三元线性方程组2.三阶行列式包括3!项,每一项都是位于不同行,不同列的三个元素的乘积,其中三项为正,三项为负.;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa,3332323222131211aabaabaabD若记333231232221131211aaaaaaaaaD或121bbb;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa,3332323222131211aabaabaabD记,3332323222131211aabaabaabD即;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa333231232221131211aaaaaaaaaD;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa,3333123221131112abaabaabaD得;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa333231232221131211aaaaaaaaaD;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa,3333123221131112abaabaabaD得;,,333323213123232221211313212111bxaxaxabxaxaxabxaxaxa.3323122221112113baabaabaaD,3333123221131112abaabaabaD.3323122221112113baabaabaaD则三元线性方程组的解为:,11DDx,22DDx.33DDx333231232221131211aaaaaaaaaD,3332323222131211aabaabaabD2-43-122-4-21D计算三阶行列式例2解按对角线法则,有D4)2()4()3(12)2(21)3(2)4()2()2(241124843264.14.094321112xx求解方程例3解方程左端1229184322xxxxD,652xx2560xx由解得3.2xx或例4解线性方程组.0,132,22321321321xxxxxxxxx解由于方程组的系数行列式111312121D1111321211111221315,0同理可得1103111221D,51013121212D,100111122213D,5故方程组的解为:,111DDx,222DDx.133DDx二阶和三阶行列式是由解二元和三元线性方程组引入的.对角线法则二阶与三阶行列式的计算.2112221122211211aaaaaaaa,312213332112322311322113312312332211aaaaaaaaaaaaaaaaaa333231232221131211aaaaaaaaa三、小结
本文标题:工科线性代数1-1
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