From 234e4931b5539ed231be9c46723df68026fd7719 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sat, 3 Jul 2021 22:48:55 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E8=A1=8C=E5=88=97=E5=BC=8F?= =?UTF-8?q?=E4=B8=8E=E7=9F=A9=E9=98=B5?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../exercise/1-determinant/determinant.pdf | Bin 98184 -> 108454 bytes .../exercise/1-determinant/determinant.tex | 170 +++++++++++++++++- .../knowledge/1-determinant/determinant.pdf | Bin 84450 -> 84771 bytes .../knowledge/1-determinant/determinant.tex | 6 +- linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 184336 -> 127531 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 89 ++++----- 6 files changed, 215 insertions(+), 50 deletions(-) diff --git a/linear-algebra/exercise/1-determinant/determinant.pdf b/linear-algebra/exercise/1-determinant/determinant.pdf index 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b/linear-algebra/exercise/1-determinant/determinant.tex @@ -2,6 +2,7 @@ % UTF8编码,ctexart现实中文 \usepackage{color} % 使用颜色 +\definecolor{orange}{RGB}{255,127,0} \usepackage{geometry} \setcounter{tocdepth}{4} \setcounter{secnumdepth}{4} @@ -167,6 +168,82 @@ $\left|\begin{array}{cc} B & * \end{array}\right|=(-1)^{mn}\vert A\vert\cdot\vert B\vert$。 +\subsubsection{爪形行列式} + +$\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & \ddots & & \\ + \vdots & & \ddots & \\ + a_{n1} & & & a_{nn} +\end{array}\right|$,$ +\left|\begin{array}{cccc} + a_{11} & & & a_{1n} \\ + a_{21} & & \begin{turn}{80}$\ddots$\end{turn} & \\ + \vdots & \begin{turn}{80}$\ddots$\end{turn} & & \\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{array}\right|$,$ +\left|\begin{array}{cccc} + a_{11} & & & a_{1n} \\ + & \ddots & & a_{2n} \\ + & & \ddots & \vdots \\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{array}\right|$, + +$ +\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + & & \begin{turn}{80}$\ddots$\end{turn} & a_{2n} \\ + & \begin{turn}{80}$\ddots$\end{turn} & & \vdots \\ + a_{n1} & & & a_{nn} +\end{array}\right|$。低阶直接进行展开,高阶需要使用递推法。 + +\subsubsection{异爪形行列式} + +每种爪形行列式都能变为三种异爪形行列式,如$\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & \ddots & & \\ + \vdots & & \ddots & \\ + a_{n1} & & & a_{nn} +\end{array}\right|$: + +$\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & \ddots & & \\ + & \ddots & \ddots & \\ + & & \ddots & a_{nn} +\end{array}\right|$,$\left|\begin{array}{cccc} + a_{11} & a_{12} & & \\ + a_{21} & \ddots & \ddots & \\ + \vdots & & \ddots & \ddots \\ + a_{n1} & & & a_{nn} +\end{array}\right|$,$\left|\begin{array}{cccc} + a_{11} & a_{12} & & \\ + a_{21} & \ddots & \ddots & \\ + & \ddots & \ddots & \ddots \\ + & & \ddots & a_{nn} +\end{array}\right|$。 + +基本方法是用斜着的数消去平的数,从而让异爪形行列式变为三角行列式。 + +\textbf{例题:}计算$\left|\begin{array}{cccc} + 1 & 1 & 1 & 1 \\ + 1 & 2 & 0 & 0 \\ + 1 & 0 & 3 & 0 \\ + 1 & 0 & 0 & 4 +\end{array}\right|$。 + +解:各行提取斜对角线数$=2\times3\times4\left|\begin{array}{cccc} + 1 & 1 & 1 & 1 \\ + \dfrac{1}{2} & 0 & 0 & 0 \\ + \dfrac{1}{3} & 0 & 1 & 0 \\ + \dfrac{1}{4} & 0 & 0 & 1 +\end{array}\right|=\left|\begin{array}{cccc} + -\dfrac{1}{12} & 0 & 0 & 0 \\ + \dfrac{1}{2} & 0 & 0 & 0 \\ + \dfrac{1}{3} & 0 & 1 & 0 \\ + \dfrac{1}{4} & 0 & 0 & 1 +\end{array}\right|=-2$。 + \subsubsection{基本行列式计算} 基本的计算方式是对角线法则计算与行列式展开两种方法。若符合基本特殊行列式的可以按照公式。 @@ -178,6 +255,8 @@ $\left|\begin{array}{cc} \item 通过行列式的对换从上往下让行列式变成上三角行列式,对角线相乘就得到结果。 \end{itemize} +这两种方式可以混合使用,直接展开可以在有足够多的0的情况下使用或阶数较低的情况下使用。\medskip + \textbf{例题:}计算$\left|\begin{array}{cccccc} a & b & 0 & \cdots & 0 & 0 \\ 0 & a & b & \cdots & 0 & 0 \\ @@ -187,7 +266,7 @@ $\left|\begin{array}{cc} b & 0 & 0 & \cdots & 0 & a \end{array}\right|$。 -直接按第一列展开: +解:直接按第一列展开: $=a(-1)^{1+1}\left|\begin{array}{ccccc} a & b & \cdots & 0 & 0 \\ @@ -195,8 +274,16 @@ $=a(-1)^{1+1}\left|\begin{array}{ccccc} \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & a & b \\ 0 & 0 & \cdots & 0 & a +\end{array}\right|+b\cdot(-1)^{n+1}\left|\begin{array}{ccccc} + b & 0 & \cdots & 0 & 0 \\ + a & b & \cdots & 0 & 0 \\ + \vdots & \vdots & \ddots & \vdots & \vdots \\ + 0 & 0 & \cdots & b & 0 \\ + 0 & 0 & \cdots & a & b \\ \end{array}\right|$ +$=a\cdot a^{n-1}+(-1)^{n+1}b\cdot b^{n-1}=a^n+(-1)^{n+1}b^n$。 + \subsection{提取公因式} 可以提取某一行或某一列的公因式。 @@ -361,10 +448,6 @@ $=3\left|\begin{array}{cccc} 0 & 0 & -\dfrac{7}{3} & 2 \\ \end{array}\right|=0$。 -% \subsection{分块行列式} - -% 当行列式左下角和右上角的矩阵为零矩阵时可以只考虑对角线矩阵的乘积值。 - \subsection{拆项} 若行列式某一行或一列是有两个值构成,则可以把其拆开,其他部分行列不变。 @@ -447,7 +530,7 @@ $ \subsection{行列加和为定值} -当行列式每一行或每一列相加都为一个固定的值,可以把第二行开始的各行都加到第一行,再提取公因式。 +当行列式每一行或每一列相加都为一个固定的值,可以把第二行开始的各行都加到第一行或列,再提取公因式,提出后第一行或第一列变为1,再依次对每行或每列进行消去,最终变成对角线行列式。 \textbf{例题:}计算$\left|\begin{array}{cccc} x & a & \cdots & a \\ @@ -498,7 +581,76 @@ $=(1+a_1+a_2+\cdots+a_n)\left|\begin{array}{cccc} a_n & 0 & \cdots & 1 \\ \end{array}\right|=(1+a_1+a_2+\cdots+a_n)$。 -\subsection{X型矩阵} +\subsection{分块行列式} + +就是分块行列式的拓展,也称为拉普拉斯展开式,当行列式左下角和右上角的矩阵为零矩阵时可以只考虑对角线矩阵的乘积值。 + +\textbf{例题:}计算$\left|\begin{array}{cccc} + a_1 & 0 & 0 & b_1 \\ + 0 & a_2 & b_2 & 0 \\ + 0 & b_3 & a_3 & 0 \\ + b_4 & 0 & 0 & a_4 +\end{array}\right|$。 + +$=-\left|\begin{array}{cccc} + a_1 & b_1 & 0 & 0 \\ + 0 & 0 & a_2 & b_2 \\ + 0 & 0 & b_3 & a_3 \\ + b_4 & a_4 & 0 & 0 +\end{array}\right|=\left|\begin{array}{cccc} + a_1 & b_1 & 0 & 0 \\ + b_4 & a_4 & 0 & 0 \\ + 0 & 0 & b_3 & a_3 \\ + 0 & 0 & a_2 & b_2 +\end{array}\right|=\left|\begin{array}{cc} + a_1 & b_1 \\ + b_4 & a_4 +\end{array}\right|\left|\begin{array}{cc} + b_3 & a_3 \\ + a_2 & b_2 +\end{array}\right|=(a_1a_4-b_1b_4)(a_2a_3-b_2b_3)$。 + +\subsection{递推法} + +当行列式一共有$n$项,很多时候都需要使用到递推法,递推法也一般与其他方法共同使用。 + +\textbf{例题:}计算$\left|\begin{array}{cccccc} + 2 & -1 & 0 & \cdots & 0 & 0 \\ + -1 & 2 & -1 & \cdots & 0 & 0 \\ + 0 & -1 & 2 & \cdots & 0 & 0 \\ + \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ + 0 & 0 & 0 & \cdots & 2 & -1 \\ + 0 & 0 & 0 & \cdots & -1 & 2 +\end{array}\right|_{n\times n}$ + +解:这是一个爪形行列式,然而无法通过斜的数数据来消去平的数据,考虑将所有列都加到第一列: + +$D_n=\left|\begin{array}{cccccc} + 1 & -1 & 0 & \cdots & 0 & 0 \\ + 0 & 2 & -1 & \cdots & 0 & 0 \\ + 0 & -1 & 2 & \cdots & 0 & 0 \\ + \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ + 0 & 0 & 0 & \cdots & 2 & -1 \\ + 1 & 0 & 0 & \cdots & -1 & 2 +\end{array}\right|=1\cdot(-1)^{1+1}\left|\begin{array}{ccccc} + 2 & -1 & \cdots & 0 & 0 \\ + -1 & 2 & \cdots & 0 & 0 \\ + \vdots & \vdots & \ddots & \vdots & \vdots \\ + 0 & 0 & \cdots & 2 & -1 \\ + 0 & 0 & \cdots & -1 & 2 +\end{array}\right|$ + +$+1\cdot(-1)^{n+1}\left|\begin{array}{cccccc} + -1 & 0 & \cdots & 0 & 0 \\ + 2 & -1 & \cdots & 0 & 0 \\ + -1 & 2 & \ddots & 0 & 0 \\ + \vdots & \vdots & \cdots & \ddots & \vdots \\ + 0 & 0 & \cdots & 2 & -1 +\end{array}\right|=D_{n-1}+1\cdot(-1)^{n+1}(-1)^{n-1}$ + +$\therefore D_n=D_{n-1}+1$。这就是本题目的递推式。$D_n=D_1+n-1=2+n-1=1+n$。 + +\textcolor{orange}{注意:}对于爪形和异爪形行列式的递推时,只能从爪尖开始消(行列式展开),不能从爪根部消,否则就不能保证爪的形状。 \textbf{例题:}计算$\left|\begin{array}{ccccccc} a & & & & & b \\ @@ -507,7 +659,7 @@ $=(1+a_1+a_2+\cdots+a_n)\left|\begin{array}{cccc} & & c & d \\ & \begin{turn}{80}$\ddots$\end{turn} & & & \ddots \\ c & & & & & d -\end{array}\right|_{2n}$。 +\end{array}\right|_{2n\times 2n}$。 解:将其$2n$行不断与$2n-1\cdots2$行对换,再将其$2n$列不断与$2n-1\cdots2$列对换,一共对换$2(2n-2)$次,一定是一个偶数: @@ -520,7 +672,7 @@ $=\left|\begin{array}{cccccccc} \vdots & \vdots & \vdots & & c & d \\ \vdots & \vdots & \vdots & \begin{turn}{80}$\ddots$\end{turn} & & & \ddots \\ 0 & 0 & c & & & & & d -\end{array}\right|_{2n}$,根据分块行列式的计算方式: +\end{array}\right|_{2n\times 2n}$,根据分块行列式的计算方式: $D_{2n}=D_2D_{2(n-1)}=(ad-bc)D_{2(n-1)}$,所以不断递推可以得到结果为$(ad-bc)^n$。 diff --git a/linear-algebra/knowledge/1-determinant/determinant.pdf b/linear-algebra/knowledge/1-determinant/determinant.pdf index 11fefa53e8dc402ee76a70280b2792f5e3e42179..cd585e58a8c5e978ab4b7b285625649f99b2fb90 100644 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zDKkuw6WO~XTHFZp>1YF1$bloDe4l1^ISI4-JK1J9aCSAy-HLkIuD|Voc~{iREyT8R zWK2uj{w>h9(jBKx{HSzFpk_vt8V&T8dg-2U${`)?;$4EYb$kjKUvLc#cKfo0VT&}= zs5A+I+Bl+4M&zh!)Q@IHC_9bWP0iE@s;5Co;et0xu4)B>Rz}D_tKXRoRqvJ?xKZvp!%3F`-VT^q}!m1f$v>6?K$CayY^=Z2EnJumIb}LA3r=)(a zx-O%4_G(n3U&1<{vW-IvPs7;Ld4FG)@<3C!ujh5u_I#Mid5vePQsInkET$-Drf&huBMuI;VB zyFY-#qcRQOUE#UAXeurDl-i=A+Q1DUK4C}uqu#2mav#Y1GnK=wI7b^!7 N7YsT1F9mU!{{ay0lN|s6 diff --git a/linear-algebra/knowledge/2-matrix/matrix.tex b/linear-algebra/knowledge/2-matrix/matrix.tex index e059b21..c796812 100644 --- a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -103,14 +103,14 @@ 设与两个矩阵都是同型矩阵$m\times n$,$A=(a_{ij})$和$B=(b_{ij})$,则其加法就是$A+B$。 -$$A+B=\left( +$A+B=\left( \begin{array}{cccc} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{m+n}+b_{m+n} \end{array} -\right)$$ +\right)$ \begin{itemize} \item $A+B=B+A$。 @@ -125,14 +125,14 @@ $$A+B=\left( 数$\lambda$与矩阵$A$的乘积记为$\lambda A$或$A\lambda$,规定: -$$\lambda A=A\lambda=\left( +$\lambda A=A\lambda=\left( \begin{array}{cccc} \lambda a_{11} & \lambda a_{12} & \cdots & \lambda a_{1n} \\ \lambda a_{21} & \lambda a_{22} & \cdots & \lambda a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda a_{m1} & \lambda a_{m2} & \cdots & \lambda a_{mn} \end{array} -\right)$$ +\right)$ 假设$A$、$B$都是$m\times n$的矩阵,$\lambda$、$\mu$为数: @@ -146,7 +146,9 @@ $$\lambda A=A\lambda=\left( \subsection{矩阵相乘} -设$A=(a_{ij})$是一个$m\times s$的矩阵,$B=(b_{ij})$是一个$s\times n$的矩阵,那么$A\times B=AB=C_{m\times n}=(c_{ij})$。即:$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}\,\text{(}i=1,2,\cdots,m;j=1,2,\cdots,n\text{)}$ +设$A=(a_{ij})$是一个$m\times s$的矩阵,$B=(b_{ij})$是一个$s\times n$的矩阵,那么$A\times B=AB=C_{m\times n}=(c_{ij})$。即:$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}\,\text{(}i=1,2,\cdots,m;j=1,2,\cdots,n\text{)}$。 + +即前一个矩阵的行乘后一个矩阵的列就得到当前元素的值。 所以按此定义一个$1\times s$行矩阵与$s\times 1$列矩阵的乘积就是一个1阶方针即一个数: @@ -197,7 +199,7 @@ $AB=A(\beta_1,\cdots,\beta_s)=(A\beta_1,\cdots,A\beta_n)$。 若$A$是$n$阶方阵,所以: -$$A^1=A\text{,}A^2=A^1A^1\text{,}\cdots\text{,}A^{k+1}=A^kA^1$$ +$A^1=A\text{,}A^2=A^1A^1\text{,}\cdots\text{,}A^{k+1}=A^kA^1$ \begin{itemize} \item $A^kA^l=A^{k+l}$。 @@ -253,6 +255,7 @@ $f(A)=A^2-A-6E=(A+2E)(A-3E)$。 \item $(A+B)^T=A^T+B^T$。 \item $(\lambda A)^T=\lambda A^T$。 \item $(AB)^T=B^TA^T$。 + \item 若$m=n$,$\vert A\vert=\vert A^T\vert$。 \end{itemize} 对称矩阵或对称阵\textcolor{violet}{\textbf{定义:}}元素以对角线为对称轴对应相等,$A=A^T$。 @@ -267,16 +270,18 @@ $f(A)=A^2-A-6E=(A+2E)(A-3E)$。 \item $\vert AB\vert=\vert A\vert\cdot\vert B\vert=\vert BA\vert$。 \end{itemize} +一般而言:$\vert A+B\vert\neq\vert A\vert+\vert B\vert$,$\vert A\vert\neq O\nRightarrow\vert A\vert\neq0$,$A\neq B\nRightarrow\vert A\vert\neq\vert B\vert$。 + 伴随矩阵或伴随阵\textcolor{violet}{\textbf{定义:}}行列式$\vert A\vert$各个元素的代数余子式$A_{ij}$转置构成的矩阵。 -$$A^*=\left( +$A^*=\left( \begin{array}{cccc} A_{11} & A_{21} & \cdots & A_{n1} \\ A_{12} & A_{22} & \cdots & A_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ A_{1n} & A_{2n} & \cdots & A_{nn} \end{array} -\right)$$ +\right)$ 其中$AA^*=A^*A=\vert A\vert E$。 @@ -292,7 +297,7 @@ $$A^*=\left( \textcolor{aqua}{\textbf{定理:}}若$AB$矩阵行列数相同,采用相同的分块法,则 -$$A=\left( +$A=\left( \begin{array}{ccc} A_{11} & \cdots & A_{1r} \\ \vdots & & \vdots \\ @@ -304,15 +309,15 @@ $$A=\left( \vdots & & \vdots \\ B_{s1} & \cdots & B_{sr} \end{array} -\right)$$ +\right)$ -$$A+B=\left( +$A+B=\left( \begin{array}{ccc} A_{11}+B_{11} & \cdots & A_{1r}+B_{1r} \\ \vdots & & \vdots \\ A_{s1}+B_{s1} & \cdots & A_{sr}+B_{sr} \end{array} -\right)\text{。}$$ +\right)\text{。}$ \textcolor{aqua}{\textbf{定理:}}设$A=\left( \begin{array}{ccc} @@ -330,7 +335,7 @@ $$A+B=\left( \textcolor{aqua}{\textbf{定理:}}若$A_{m\times l}$,$B_{l\times n}$,采用相同的分块法,则 -$$A=\left( +$A=\left( \begin{array}{ccc} A_{11} & \cdots & A_{1t} \\ \vdots & & \vdots \\ @@ -342,15 +347,15 @@ $$A=\left( \vdots & & \vdots \\ B_{t1} & \cdots & B_{sr} \end{array} -\right)$$ +\right)$ -$$AB=\left( +$AB=\left( \begin{array}{ccc} C_{11} & \cdots & C_{1r} \\ \vdots & & \vdots \\ C_{s1} & \cdots & C_{sr} \end{array} -\right)\text{,}C_{ij}=\sum\limits_{k=1}^tA_{ik}B_{kj}\text{。}$$ +\right)\text{,}C_{ij}=\sum\limits_{k=1}^tA_{ik}B_{kj}\text{。}$ \textcolor{aqua}{\textbf{定理:}}设$A=\left( \begin{array}{ccc} @@ -404,28 +409,30 @@ $$AB=\left( 若对于$A_{m\times s}$与$B_{s\times n}$的乘积矩阵$AB=C=(c_{ij})_{m\times n}$,若将$A$按行分为$m$块,$B$按列分为$n$块,则有: -$$AB=\left( +$AB=\left( \begin{array}{c} a_1^T \\ a_2^T \\ \vdots \\ a_{m}^T \end{array} -\right)(b_1,b_2,\cdots,b_n)=\left( +\right)(b_1,b_2,\cdots,b_n)$ + +$=\left( \begin{array}{cccc} a_1^Tb_1 & a_1^Tb_2 & \cdots & a_1^Tb_n \\ a_2^Tb_1 & a_2^Tb_2 & \cdots & a_2^Tb_n \\ \vdots & \vdots & \ddots & \vdots \\ a_{m}^Tb_1 & a_{m}^Tb_2 & \cdots & a_{m}^Tb_n \end{array} -\right)=(c_{ij})_{m\times n}\text{。}$$ +\right)=(c_{ij})_{m\times n}\text{。}$ -$$c_{ij}=a_i^Tb_j=(a_{i1},a_{i2},\cdots,a_{is})\left(\begin{array}{c} +其中:$c_{ij}=a_i^Tb_j=(a_{i1},a_{i2},\cdots,a_{is})\left(\begin{array}{c} b_{1j} \\ b_{2j} \\ \vdots \\ b_{sj} -\end{array}\right)=\sum\limits_{k=1}^s=a_{ik}b_{kj}\text{。}$$ +\end{array}\right)=\sum\limits_{k=1}^s=a_{ik}b_{kj}\text{。}$ \textcolor{aqua}{\textbf{定理:}}$A=O$的充要条件是$A^TA=O$。 @@ -433,7 +440,7 @@ $$c_{ij}=a_i^Tb_j=(a_{i1},a_{i2},\cdots,a_{is})\left(\begin{array}{c} 设$A=(a_{ij})_{m\times n}$,将$A$按列分块为$A=(a_1,a_2,\cdots,a_n)$,则 -$$A^TA=\left( +$A^TA=\left( \begin{array}{c} a_1^T \\ a_2^T \\ @@ -447,7 +454,7 @@ $$A^TA=\left( \vdots & \vdots & \ddots & \vdots \\ a_{m}^Ta_1 & a_{m}^Ta_2 & \cdots & a_{m}^Ta_n \end{array} -\right)\text{。}$$ +\right)\text{。}$ 所以$A^TA$的元为$a^T_ia_j$,又$\because A^TA=O$,$\therefore a^T_ia_j=0$($i,j=1,2,\cdots n$)。 @@ -554,34 +561,34 @@ $\therefore A=O$。 矩阵乘法实际上就是线性方程组的线性变换,将一个变量关于另一个变量的关系式代入原方程组,得到与另一个变量的关系。 -$$\begin{cases} +$\begin{cases} y_1=a_{11}x_1+a_{12}x_2+\cdots+a_{1s}x_s \\ \cdots \\ y_m=a_{m1}x_1+a_{m2}x_2+\cdots+a_{ms}x_s -\end{cases}\text{,}\begin{cases} +\end{cases}\begin{cases} x_1=b_{11}t_1+b_{12}t_2+\cdots+b_{1n}t_n \\ \cdots \\ x_s=b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n -\end{cases}\text{。}$$ +\end{cases}$ 原本是线性方程分别是$y$与$x$和$x$与$t$的关系式,而如果将$t$关于$x$的关系式代入$x$关于$y$的关系式中,就会得到$t$关于$y$的关系式: -$$\begin{cases} - y_1=a_{11}(b_{11}t_1+b_{12}t_2+\cdots+b_{1n}t_n)+\cdots+a_{1s}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) \\ +$\begin{cases} + y_1=a_{11}(b_{11}t_1+\cdots+b_{1n}t_n)+\cdots+a_{1s}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) \\ \cdots \\ - y_m=a_{m1}(b_{11}t_1+b_{12}t_2+\cdots+b_{1n}t_n)+\cdots+a_{ms}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) -\end{cases}$$ + y_m=a_{m1}(b_{11}t_1+\cdots+b_{1n}t_n)+\cdots+a_{ms}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) +\end{cases}$ -$$=\begin{cases} +$=\begin{cases} y_1=(a_{11}b_{11}+\cdots+a_{1s}b_{s1})t_1+\cdots+(a_{11}b_{1n}+\cdots+a_{1s}b_{sn})t_n \\ \cdots \\ y_m=(a_{m1}b_{11}+\cdots+a_{ms}b_{s1})t_1+\cdots+(a_{m1}b_{1n}+\cdots+a_{ms}b_{sn})t_m -\end{cases}$$ +\end{cases}$ 这可以看作上面两个线性方程组相乘,也可以将线性方程组表示为矩阵,进行相乘就得到乘积,从而了解矩阵乘积与线性方程组的关系: -$$\left(\begin{array}{ccc} +$\left(\begin{array}{ccc} a_{11} & \cdots & a_{1s} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{ms} @@ -589,13 +596,13 @@ $$\left(\begin{array}{ccc} b_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ b_{s1} & \cdots & b_{sn} -\end{array}\right)_{s\times n}$$ +\end{array}\right)_{s\times n}$ -$$=\left(\begin{array}{ccc} +$=\left(\begin{array}{ccc} a_{11}b_{11}+\cdots+a_{1s}b_{s1} & \cdots & a_{11}b_{1n}+\cdots+a_{1s}b_{sn} \\ \vdots & \ddots & \vdots \\ a_{m1}b_{11}+\cdots+a_{ms}b_{s1} & \cdots & a_{m1}b_{1n}+\cdots+a_{ms}b_{sn} -\end{array}\right)_{m\times n}\text{。}$$ +\end{array}\right)_{m\times n}\text{。}$ \subsection{线性方程组的解} @@ -636,7 +643,7 @@ $$=\left(\begin{array}{ccc} 按乘积表示为$A_{m\times n}x_{n\times 1}=b_{m\times 1}$,然后将$A$按列分块,$x$按行分块: -$$(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} +$(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} x_1 \\ x_2 \\ \vdots \\ @@ -656,7 +663,7 @@ $$(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} b_2 \\ \vdots \\ b_m -\end{array}\right)\text{。}$$ +\end{array}\right)\text{。}$ 这三种都是解的表示方法。 @@ -795,7 +802,7 @@ $(A\vdots B)\overset{r}{\thicksim}(E\vdots A^{-1})$,$\left(\begin{array}{c} 解:因为$AX=B$,所以左乘$A^{-1}$:$A^{-1}AX=EX=A^{-1}B$,增广矩阵行变换: -$$(A,B)=\left(\begin{array}{ccccc} +$(A,B)=\left(\begin{array}{ccccc} 2 & 1 & -3 & 1 & -1 \\ 1 & 2 & -2 & 2 & 0 \\ -1 & 3 & 2 & -2 & 5 @@ -803,7 +810,7 @@ $$(A,B)=\left(\begin{array}{ccccc} 1 & 2 & -2 & 2 & 0 \\ 0 & -3 & 1 & -3 & -1 \\ 0 & 5 & 0 & 0 & 5 -\end{array}\right)$$ +\end{array}\right)$ $\thicksim\left(\begin{array}{ccccc} 1 & 2 & -2 & 2 & 0 \\ @@ -821,4 +828,6 @@ $\thicksim\left(\begin{array}{ccccc} \section{矩阵秩} +秩的本质就是组成矩阵的线性无关的向量个数。 + \end{document} \ No 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