From dc8cb96fac4493f56dda42a1c46689128ff010c7 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 5 Jul 2021 23:27:26 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=9F=A9=E9=98=B5=E5=92=8C?= =?UTF-8?q?=E5=90=91=E9=87=8F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .gitignore | 1 + linear-algebra/exercise/2-matrix/matrix.pdf | Bin 57020 -> 74351 bytes linear-algebra/exercise/2-matrix/matrix.tex | 184 +++- linear-algebra/exercise/3-vector/vector.pdf | Bin 0 -> 25359 bytes linear-algebra/exercise/3-vector/vector.tex | 44 + linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 132069 -> 134648 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 74 +- .../knowledge/2-matrix/matrix.tex.bak | 930 ++++++++++++++++++ linear-algebra/knowledge/3-vector/vector.pdf | Bin 0 -> 38126 bytes linear-algebra/knowledge/3-vector/vector.tex | 69 ++ 10 files changed, 1286 insertions(+), 16 deletions(-) create mode 100644 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+ 0 & 2 & -4 +\end{array}\right)$,$B=\left(\begin{array}{ccc} + 1 & 0 & 1 \\ + 0 & -1 & 2 \\ + 0 & 0 & 0 +\end{array}\right)$,当$A\thicksim B$时,求$P$使得$PA=B$。. + +解:目标是将$A$变为$B$,所以第一步将第一列的第二行的-1变为0。即将第一行加到第二行。 + +左乘$E_{21}(1)A=\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 0 & 0 & 1 +\end{array}\right)\left(\begin{array}{ccc} + 1 & 0 & 1 \\ + -1 & -1 & 1 \\ + 0 & 2 & -4 +\end{array}\right)=\left(\begin{array}{ccc} + 1 & 0 & 1 \\ + 0 & -1 & 2 \\ + 0 & 2 & -4 +\end{array}\right)=C$。\medskip + +然后对第二列进行消,首先将第三行加上第二行的两倍。 + +$E_{32}(2)C=\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 0 & 2 & 1 +\end{array}\right)\left(\begin{array}{ccc} + 1 & 0 & 1 \\ + 0 & -1 & 2 \\ + 0 & 2 & -4 +\end{array}\right)=\left(\begin{array}{ccc} + 1 & 0 & 1 \\ + 1 & -1 & 2 \\ + 0 & 0 & 0 +\end{array}\right)=B$。\medskip + +$\therefore E_{32}(2)E_{21}(1)A=B$。 + +$P=E_{32}(2)E_{21}(1)=\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 2 & 1 +\end{array}\right)\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 0 & 0 & 1 +\end{array}\right)=\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 1 & 1 & 0 \\ + 2 & 2 & 1 +\end{array}\right)$。 + \section{逆矩阵} \subsection{定义法} @@ -170,8 +238,20 @@ $\because A+B=A(E+A^{-1}B)=A(B^{-1}+A^{-1})B$。 $\therefore (A+B)^{-1}=B^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}$。 +\subsection{初等变换} + +$\left[A\vdots B\right]\overset{r}{\thicksim}\left[E\vdots A^{-1}\right]$,$\left[\begin{array}{c} + A \\ + B +\end{array}\right]\overset{c}{\thicksim}\left[\begin{array}{c} + E \\ + A^{-1} +\end{array}\right]$。 + \subsection{分块矩阵} +基于拉普拉斯展开式。 + 对于一些分块矩阵的逆,若$A$,$B$都可逆,则:$\left[\begin{array}{cc} A & O \\ O & B @@ -184,6 +264,108 @@ $\therefore (A+B)^{-1}=B^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}$。 \end{array}\right]^{-1}=\left[\begin{array}{cc} O & B^{-1} \\ A^{-1} & O -\end{array}\right]$。 +\end{array}\right]$。\medskip + +\textbf{例题:}已知$A=\left(\begin{array}{cc} + B & O \\ + D & C +\end{array}\right)$,其中$B$为$r\times r$可逆矩阵,$C$为$s\times s$可逆矩阵,求$A^{-1}$。 + +解:$\because\vert A\vert=\left|\begin{array}{cc} + B & O \\ + D & C +\end{array}\right|=\vert B\vert\vert C\vert\neq0$,所以$A$可逆,设$A^{-1}=\left(\begin{array}{cc} + X & Y \\ + Z & W +\end{array}\right)$。 + +$AA^{-1}=\left(\begin{array}{cc} + E_r & O \\ + O & E_s +\end{array}\right)=E_{r+s}$。即$\left(\begin{array}{cc} + BX & BY \\ + DX+CZ & DY+CW +\end{array}\right)=E_{r+s}$。 + +$\therefore\left\{\begin{array}{l} + BX=E \\ + BY=O \\ + DX+CZ=O \\ + DY+CW=E +\end{array}\right.$,$\left\{\begin{array}{ll} + B^{-1}BX=B^{-1}, & X=B^{-1}\\ + B^{-1}BY=O, & Y=O \\ + CZ=-DX=-DB^{-1}, & Z=-C^{-1}DB^{-1} \\ + CW=E, & W=C^{-1} +\end{array}\right.$。 + +$\therefore A^{-1}=\left(\begin{array}{cc} + B^{-1} & O \\ + -C^{-1}DB^{-1} & C^{-1} +\end{array}\right)$。\medskip + +当分块矩阵为三角矩阵时,对角线为原方块矩阵的逆矩阵,非0的一角为原矩阵,再左乘同行的逆矩阵,右乘同列的逆矩阵。\medskip + +$\therefore A=\left(\begin{array}{cc} + B & D \\ + O & C +\end{array}\right)$,$A^{-1}=\left(\begin{array}{cc} + B^{-1} & -B^{-1}DC^{-1} \\ + O & C^{-1} +\end{array}\right)$。\medskip + +当分块矩阵为副对角矩阵时,对角线为对角方块矩阵的逆矩阵,非0的一角为原矩阵,再左乘同行的逆矩阵,右乘同列的逆矩阵。\medskip + +$\therefore A=\left(\begin{array}{cc} + O & B \\ + C & D +\end{array}\right)$,$A^{-1}=\left(\begin{array}{cc} + -C^{-1}DB^{-1} & C^{-1} \\ + B^{-1} & O +\end{array}\right)$。\medskip + +$\therefore A=\left(\begin{array}{cc} + D & B \\ + C & O +\end{array}\right)$,$A^{-1}=\left(\begin{array}{cc} + O & C^{-1} \\ + B^{-1} & -C^{-1}DB^{-1} +\end{array}\right)$。\medskip + +$A=\left(\begin{array}{ccc} + A_1 \\ + & \ddots \\ + & & A_n +\end{array}\right)$,$A^{-1}=\left(\begin{array}{ccc} + A_1^{-1} \\ + & \ddots \\ + & & A_n^{-1} +\end{array}\right)$。\medskip + +$A=\left(\begin{array}{ccc} + & & A_1 \\ + & \ddots \\ + A_n +\end{array}\right)$,$A^{-1}=\left(\begin{array}{ccc} + & & A_n^{-1} \\ + & \begin{turn}{80}$\ddots$\end{turn} \\ + A_1^{-1} +\end{array}\right)$。 + +\section{矩阵方程} + +含有未知矩阵的方程就是矩阵方程,需要将方程进行恒等变形,化为$AX=B$、$XA=B$或$AXB=C$的形式。 + +若$A$、$B$可逆,且可以分别得到$X=A^{-1}B$,$X=BA^{-1}$,$X=A^{-1}CB^{-1}$。 + +\textbf{例题:}设3阶方阵$A$,$B$满足$A^{-1}BA=6A+BA$,且$A=\left(\begin{array}{ccc} + \dfrac{1}{3} & 0 & 0 \\ + 0 & \dfrac{1}{4} & 0 \\ + 0 & 0 & \dfrac{1}{5} +\end{array}\right)$,求$B$。 + +解:$A^{-1}BA=(6E+B)A$,$A^{-1}B=6E+B$,$A^{-1}B-B=6E$,$(A^{-1}-E)B=6E$。 + +$\therefore B=6(A^{-1}-E)^{-1}$。 \end{document} diff --git a/linear-algebra/exercise/3-vector/vector.pdf b/linear-algebra/exercise/3-vector/vector.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c9e6b5c93660afe0eed93b35a14977fad92659a1 GIT binary patch literal 25359 zcma&NW0Ymdx;31Ywr$(CZQHhOyV9(*D{Z^dwr$(~s?Rz1z1`pKaYoM{J7TY~N37WM zVLTB*A}=gT!$8XdNwR-^@eIkpj8BhmXJ`q@%}pn2VeM?}XFU8D|ILg9X>ujA7y)rT za;}QFsdZrscOP5?!|02*6F5h_7h=WK7fqAc-QTJ%8I3EVCqL$=4snPm3@jms^QDvD zjn|JwWLeGS`P$;2(DnA!qc$#$w(`|@sihfw$g=<&snVJn2z@()u-578P4B*asI~}N zsN@DMrdPG6#0Sf!A77m6rsA(gTMNfUU7K0o9P$l@Ax&(J|MN$G_x@mpk?EgM#KQa+ ziUyRWVv!i2yY8qQGjRu(W01&JO^QSUlUoQLK(X8AFkr!nZQs2!JsQ#}QN)zL-yR={ 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a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -752,13 +752,17 @@ $(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} \item 计算$A^{-1}=\dfrac{1}{\vert A\vert}A^*$。 \end{enumerate} +\subsubsection{初等变换} + +可以利用初等变换来求逆矩阵。 + \section{矩阵初等变换} -求逆矩阵可以使用伴随矩阵来求,但是只针对三阶以及以下的矩阵,若阶数过高则会十分困难。可以使用矩阵初等变换来实现求逆矩阵。且初等变换还可以用来求线性方程组的解。 +可以使用矩阵初等变换来实现求逆矩阵。且初等变换还可以用来求线性方程组的解。 \subsection{初等变换} -矩阵的三种初等行变换: +矩阵的三种初等行变换,互换、倍乘、倍加: \begin{enumerate} \item 对换两行(对换$ij$两行,记为$r_i\leftrightarrow r_j$)。 @@ -811,39 +815,79 @@ $(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} \subsection{初等变换性质} -\textcolor{aqua}{\textbf{定理:}}设$AB$都是$m\times n$矩阵:\begin{enumerate} +\textcolor{aqua}{\textbf{定理:}}设$AB$都是$m\times n$矩阵,初等变换与矩阵乘积关系:\begin{enumerate} \item $A\overset{r}{\thicksim}B$的充要条件是存在$m$阶可逆矩阵$P$,使得$PA=B$。 \item $A\overset{c}{\thicksim}B$的充要条件是存在$n$阶可逆矩阵$Q$,使得$AQ=B$。 \item $A\thicksim B$的充要条件是存在$m$阶可逆矩阵$P$和$n$阶可逆矩阵$Q$,使得$PAQ=B$。 \end{enumerate} -初等矩阵\textcolor{violet}{\textbf{定义:}}由单位矩阵$E$经过一次初等变换得到的矩阵。 +初等变换具有如下性质: -初等矩阵包括:\begin{enumerate} +\begin{itemize} + \item 设$A$是一个$m\times n$矩阵,对$A$进行一次初等行变换,相当于在$A$左乘对应$m$阶初等矩阵;对$A$进行一次列变换,相当于在$A$右乘对应$n$阶初等矩阵。 + \item 方阵$A$可逆的充分必要条件是存在有限个初等矩阵$P_i$使得$A=\prod\limits_{i=1}^nP_i$。 + \item 可逆方阵$A$一定可以通过有限次初等变换化为同阶单位矩阵$E$。 + \item 方阵$A$可逆的充要条件是$A\overset{r}{\thicksim}E$。(即$A$方阵所代表的线性方程组能通过初等计算得到最后的解) +\end{itemize} + +对于$A_{m\times n}$进行初等变换:\begin{enumerate} \item 第$ij$行对换:$E_m(ij)A$,第$ij$列变换:$AE_n(ij)$。 \item 数$k$乘第$i$行:$E_m(i(k))A$,数$k$乘第$i$列:$AE_n(i(k))$。 \item 数$k$乘第$j$行加到$i$行:$E_m(ij(k))A$,数$k$乘第$j$列加到$i$列:$AE_n(ij(k))$。 \end{enumerate} -\textcolor{aqua}{\textbf{定理:}}设$A$是一个$m\times n$矩阵,对$A$进行一次初等行变换,相当于在$A$左乘对应$m$阶初等矩阵;对$A$进行一次列变换,相当于在$A$右乘对应$n$阶初等矩阵。 +已知$A$经过无数次初等变换就能变成单位矩阵,即通过乘无数个初等矩阵就可以变成单位矩阵,那么这些初等矩阵是什么呢?\medskip -\textcolor{aqua}{\textbf{定理:}}方阵$A$可逆的充分必要条件是存在有限个初等矩阵$P_i$使得$A=\prod\limits_{i=1}^nP_i$。 +例如$A=\left(\begin{array}{cc} + 1 & 2 \\ + 2 & 3 +\end{array}\right)$要变成$\left(\begin{array}{cc} + 1 & 2 \\ + 0 & -1 +\end{array}\right)$,就需要将第一排的数据乘-2加到第二排。 -\textcolor{aqua}{\textbf{定理:}}方阵$A$可逆的充要条件是$A\overset{r}{\thicksim}E$。(即$A$方阵所代表的线性方程组能通过初等计算得到最后的解) +即按照初等矩阵的表示方法就是$E_{21}(-2)$,然后这个初等矩阵就是对单位矩阵进行同样变换。 + +即$E_{21}(-2)$就是将初等矩阵第一排的数据乘-2加到第二排,得到$E_{21}(-2)=\left(\begin{array}{cc} + 1 & 0 \\ + -2 & 1 +\end{array}\right)$,而行变换$\left(\begin{array}{cc} + 1 & 0 \\ + -2 & 1 +\end{array}\right)\left(\begin{array}{cc} + 1 & 2 \\ + 2 & 3 +\end{array}\right)=\left(\begin{array}{cc} + 1 & 2 \\ + 0 & -1 +\end{array}\right)$,果然就得到目标结果。 + +从而对一个矩阵进行初等行变换就是左乘一个进行同样行变换的初等矩阵,列变换同理。 + +\subsection{初等矩阵性质} + +初等矩阵\textcolor{violet}{\textbf{定义:}}由单位矩阵$E$经过一次初等变换得到的矩阵。所以初等矩阵都是方阵。 + +\begin{itemize} + \item 初等矩阵的转置也是初等矩阵。 + \item 对初等矩阵进行行或列变换,$\vert E_{ij}\vert=-1$,对其求逆:$E_{ij}^{-1}=E_{ij}$。 + \item 对初等矩阵$i$行乘$k$,$\vert E_i(k)\vert=k$,对其求逆:$E_i(k)^{-1}=E_i\left(\dfrac{1}{k}\right)$。 + \item 对初等矩阵第$j$行乘$k$加到$i$行,$\vert E_{ij}(k)\vert=1$,对其求逆:$E_{ij}(k)^{-1}=E_{ij}(-k)$。 +\end{itemize} \subsection{初等行变换求逆} -已知$A^{-1}=\dfrac{A^*}{\vert A\vert}$,但是伴随矩阵计算非常麻烦,并且若矩阵在三阶以上计算就很难办到,所以还有一种方法,就是若该矩阵$A$是可逆矩阵,就将$AX=B$的增广矩阵$(A,B)$化为最简形矩阵,从而得到方程解。\medskip +若该矩阵$A$是可逆矩阵,就将$AX=B$的增广矩阵$(A,B)$化为最简形矩阵,从而得到方程解。\medskip -$(A\vdots B)\overset{r}{\thicksim}(E\vdots A^{-1})$,$\left(\begin{array}{c} +$\because P_i\cdots P_2P_1A=E$,$P_i\cdots P_2P_1E=A^{-1}$。 + +$\left[A\vdots B\right]\overset{r}{\thicksim}\left[E\vdots A^{-1}\right]$,$\left[\begin{array}{c} A \\ - \cdots \\ B -\end{array}\right)\overset{c}{\thicksim}\left(\begin{array}{c} +\end{array}\right]\overset{c}{\thicksim}\left[\begin{array}{c} E \\ - \cdots \\ A^{-1} -\end{array}\right)$。 +\end{array}\right]$。 \textbf{例题:}求解矩阵方程$AX=B$,$A=\left(\begin{array}{ccc} 2 & 1 & -3 \\ @@ -853,7 +897,7 @@ $(A\vdots B)\overset{r}{\thicksim}(E\vdots A^{-1})$,$\left(\begin{array}{c} 1 & -1 \\ 2 & 0 \\ -2 & 5 -\end{array}\right)$。 +\end{array}\right)$。\medskip 解:因为$AX=B$,所以左乘$A^{-1}$:$A^{-1}AX=EX=A^{-1}B$,增广矩阵行变换: diff --git a/linear-algebra/knowledge/2-matrix/matrix.tex.bak b/linear-algebra/knowledge/2-matrix/matrix.tex.bak new file mode 100644 index 0000000..2207bca --- /dev/null +++ b/linear-algebra/knowledge/2-matrix/matrix.tex.bak @@ -0,0 +1,930 @@ +\documentclass[UTF8, 12pt]{ctexart} +% UTF8编码,ctexart现实中文 +\usepackage{color} +% 使用颜色 +\definecolor{orange}{RGB}{255,127,0} +\definecolor{violet}{RGB}{192,0,255} +\definecolor{aqua}{RGB}{0,255,255} +\usepackage{geometry} +\setcounter{tocdepth}{4} +\setcounter{secnumdepth}{4} +% 设置四级目录与标题 +\geometry{papersize={21cm,29.7cm}} +% 默认大小为A4 +\geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} +% 默认页边距为1英尺与1.25英尺 +\usepackage{indentfirst} +\setlength{\parindent}{2.45em} +% 首行缩进2个中文字符 +\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\usepackage{amssymb} +% 因为所以 +\usepackage{amsmath} +% 数学公式 +\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} +% 超链接 +\usepackage{multicol} +% 分栏 +\usepackage{arydshln} +\setlength{\dashlinegap}{1pt} +\setlength{\dashlinedash}{1pt} +% 阶梯矩阵的虚线 +\author{Didnelpsun} +\title{矩阵} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} + +矩阵本质是一个表格。 + +\section{矩阵定义} + +\textcolor{violet}{\textbf{定义:}}$m\times n$矩阵是由$m\times n$个数$a_{ij}$(元素)排成的$m$行$n$列的数表。 + +元素是实数的矩阵称为\textbf{实矩阵},元素是复数的矩阵是\textbf{复矩阵}。 + +行数列数都为$n$的就是\textbf{$n$阶矩阵}或\textbf{方阵},记为$A_n$。 + +行矩阵或行向量\textcolor{violet}{\textbf{定义:}}只有一行的矩阵$A=(a_1a_2\cdots a_n)$。 + +列矩阵或列向量\textcolor{violet}{\textbf{定义:}}只有一列的矩阵$B= +\left(\begin{array}{c} + b_1 \\ + b_2 \\ + \cdots \\ + b_m +\end{array}\right)$。 + +同型矩阵\textcolor{violet}{\textbf{定义:}}两个矩阵行数、列数相等。 + +相等矩阵\textcolor{violet}{\textbf{定义:}}是同型矩阵,且对应元素相等的矩阵。记为$A=B$。 + +零矩阵\textcolor{violet}{\textbf{定义:}}元素都是零的矩阵,记为$O$,但是不同型的零矩阵不相等。 + +\begin{multicols}{2} + + + 对角矩阵或对角阵\textcolor{violet}{\textbf{定义:}}从左上角到右下角的直线(对角线)以外的元素都是0的矩阵,记为$\varLambda=\textrm{diag}(\lambda_1,\lambda_2,\cdots,\lambda_n)$。 + + $\varLambda=\left( + \begin{array}{cccc} + \lambda_1 & 0 & \cdots & 0 \\ + 0 & \lambda_2 & \cdots & 0 \\ + \vdots & \vdots & \vdots & \vdots \\ + 0 & 0 & \cdots & \lambda_n + \end{array} + \right)$ + + 单位矩阵或单位阵\textcolor{violet}{\textbf{定义:}}$\lambda_1=\lambda_2=\cdots=\lambda_n=1$的对角矩阵,记为$E$。这种线性变换叫做恒等变换,$AE=A$。 \medskip + + $E=\left( + \begin{array}{cccc} + 1 & 0 & \cdots & 0 \\ + 0 & 1 & \cdots & 0 \\ + \vdots & \vdots & \vdots & \vdots \\ + 0 & 0 & \cdots & 1 + \end{array} + \right)$ + +\end{multicols} + +\section{矩阵运算} + +\subsection{矩阵加法减法} + +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$A+B$。 + +$A+B=\left( + \begin{array}{cccc} + a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ + a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ + \vdots & \vdots & \vdots & \vdots \\ + a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{m+n}+b_{m+n} + \end{array} +\right)$ + +\begin{itemize} + \item $A+B=B+A$。 + \item $(A+B)+C=A+(B+C)$。 +\end{itemize} + +若$-A=(-a_{ij})$,则$-A$是$A$的负矩阵,$A+(-A)=O$。 + +从而矩阵的减法为$A-B=A+(-B)$。 + +\subsection{数乘矩阵} + +数$\lambda$与矩阵$A$的乘积记为$\lambda A$或$A\lambda$,规定:\medskip + +$\lambda A=A\lambda=\left( + \begin{array}{cccc} + \lambda a_{11} & \lambda a_{12} & \cdots & \lambda a_{1n} \\ + \lambda a_{21} & \lambda a_{22} & \cdots & \lambda a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + \lambda a_{m1} & \lambda a_{m2} & \cdots & \lambda a_{mn} + \end{array} +\right)$ \medskip + +假设$A$、$B$都是$m\times n$的矩阵,$\lambda$、$\mu$为数: + +\begin{itemize} + \item $(\lambda\mu)A=\lambda(\mu A)$。 + \item $(\lambda+\mu)A=\lambda A+\mu A$。 + \item $\lambda(A+B)=\lambda A+\lambda B$。 +\end{itemize} + +矩阵加法与数乘矩阵都是矩阵的线性运算。 + +\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:$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}\,\text{(}i=1,2,\cdots,m;j=1,2,\cdots,n\text{)}$。 + +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: + +$(a_{i1},a_{i2},\cdots,a_{is})\left( + \begin{array}{c} + b_{1j} \\ + b_{2j} \\ + \cdots \\ + b_{sj} + \end{array} +\right)=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}=c_{ij}$。\medskip + +从而$AB=C$的$c_{ij}$就是$A$的第$i$行与$B$的$j$列的乘积。 + +当$A$左边乘$P$为$PA$,称为\textbf{左乘}$P$,若右边乘$P$为$AP$,则称为\textbf{右乘}$P$。 + +\textcolor{orange}{注意:}只有左矩阵的列数等于右矩阵的行数才能相乘。 + +只有$AB$都是方阵的时候才能$AB$与$BA$。 + +矩阵的左乘与右乘不一定相等,即$AB\neq BA$。 + +\textcolor{violet}{\textbf{定义:}}若方阵$AB$乘积满足$AB=BA$,则表示其是\textbf{可交换}的。 + +$A\neq O$,$B\neq O$,但是不能推出$AB\neq O$或$BA\neq O$。 + +$AB=O$不能推出$A=O$或$B=O$。 + +$A(X-Y)=O$当$A\neq O$也不能推出$X=Y$。 + +\begin{itemize} + \item $(AB)C=A(BC)$。 + \item $\lambda(AB)=(\lambda A)B=A(\lambda B)$。 + \item $A(B+C)=AB+AC$。 + \item $(B+C)A=BA+CA$。 + \item $EA=AE=A$。 +\end{itemize} + +$\lambda E$称为\textbf{纯量阵},$(\lambda E_n)A_n=\lambda A_n=A_n(\lambda E_n)$。 + +若$A_{m\times s}$,$B_{s\times n}=(\beta_1,\cdots,\beta_s)$,其中$\beta$为$n$行的列矩阵,则: + +$AB=A(\beta_1,\cdots,\beta_s)=(A\beta_1,\cdots,A\beta_n)$。 + +\subsection{矩阵幂} + +只有方阵才能连乘,从而只有方阵才有幂。 + +若$A$是$n$阶方阵,所以: + +$A^1=A\text{,}A^2=A^1A^1\text{,}\cdots\text{,}A^{k+1}=A^kA^1$ + +\begin{itemize} + \item $A^kA^l=A^{k+l}$。 + \item $(A^k)^l=A^{kl}$。 +\end{itemize} + +因为矩阵乘法一般不满足交换率,所以$(AB)^k\neq A^kB^k$。只有$AB$可交换时才相等。 + +若$A\neq 0$不能推出$A^k\neq 0$,如:\medskip + +$A=\left( + \begin{array}{cc} + 0 & 2 \\ + 0 & 0 + \end{array} +\right)\neq 0$。$A^2=\left( + \begin{array}{cc} + 0 & 2 \\ + 0 & 0 + \end{array} +\right)\left( + \begin{array}{cc} + 0 & 2 \\ + 0 & 0 + \end{array} +\right)=\left( + \begin{array}{cc} + 0 & 0 \\ + 0 & 0 + \end{array} +\right)=O$。\medskip + +$A=\left( + \begin{array}{ccc} + 0 & 1 & 1 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 + \end{array} +\right)$,$A^3=O$。\medskip + +矩阵幂可以同普通多项式进行处理。 + +如$f(x)=a_nx^n+\cdots+a_1x+n$,对于$A$就是$f(A)=a_nA^n+\cdots+a_1A+a_nE$。 + +$f(A)=A^2-A-6E=(A+2E)(A-3E)$。 + +\subsection{矩阵转置} + +把矩阵$A$的行换成同序数的列就得到一个新矩阵,就是$A$的转置矩阵$A^T$。若$A$为$m\times n$,则$A^T$为$n\times m$。 + +\begin{itemize} + \item $(A^T)^T=A$。 + \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:}}矩阵$A$是方阵,且元素以对角线为对称轴对应相等,$A=A^T$。 + +反对称矩阵\textcolor{violet}{\textbf{定义:}}矩阵$A$是方阵,且满足$-A=A^T$。即$\left\{\begin{array}{l} + a_{ij}=-a_{ji},i\neq j \\ + a_{ii}=0 +\end{array}\right.$。 + +正交矩阵\textcolor{violet}{\textbf{定义:}}矩阵$A$是方阵,且满足$A^TA=E$。 + +\subsection{方阵行列式} + +由$n$阶方阵$A$的元素所构成的行列式称为矩阵$A$的行列式,记为$\textrm{det}\,A$或$\vert A\vert$。 + +\begin{itemize} + \item $\vert A^T\vert=\vert A\vert$。 + \item $\vert\lambda A\vert=\lambda^n\vert A\vert$。 + \item $\vert AB\vert=\vert A\vert\cdot\vert B\vert=\vert BA\vert$。 +\end{itemize} + +一般而言:$\vert A+B\vert\neq\vert A\vert+\vert B\vert$,$\vert A\vert\neq O\nRightarrow\vert A\vert\neq0$,$A\neq B\nRightarrow\vert A\vert\neq\vert B\vert$。 + +\subsection{伴随矩阵} + +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。 + +$A^*=\left( + \begin{array}{cccc} + A_{11} & A_{21} & \cdots & A_{n1} \\ + A_{12} & A_{22} & \cdots & A_{n2} \\ + \vdots & \vdots & \ddots & \vdots \\ + A_{1n} & A_{2n} & \cdots & A_{nn} + \end{array} +\right)$ + +\begin{itemize} + \item 任何方阵都有伴随矩阵,其中$AA^*=A^*A=\vert A\vert E$。 + \item $A^*=\vert A\vert A^{-1}$,$A^{-1}=\dfrac{1}{\vert A\vert}A^*$,$A=\vert A\vert(A^*)^{-1}$。 + \item $\vert A^*\vert=\vert A\vert^{n-1}$,$(kA)(kA)^*=\vert kA\vert E$,$A^T(A^T)^*=\vert A^T\vert E$,$A^{-1}(A^{-1})^*=\vert A^{-1}\vert E$,$A^*(A^*)^*=\vert A^*\vert E$。 + \item $(A^T)^*=(A^*)^T$,$(A^{-1})^*=(A^*)^{-1}$,$(AB)^*=B^*A^*$,$(A^*)^*=\vert A\vert^{n-2}A$。 +\end{itemize} + +\textbf{例题:}假设$A$为$n$阶方阵,求$\vert A^*\vert$与$(A^*)^*$。 + +解:$\because AA^*=A^*A=\vert A\vert E$,$\therefore A^*(A^*)^*=\vert A^*\vert E$。 + +$(A^*)^{-1}A^*(A^*)^*=(A^*)^*=(A^*)^{-1}\vert A^*\vert E$,又$AA^*=\vert A\vert E$,$\vert AA^*\vert=\vert\vert A\vert E\vert$, + +$\therefore\vert A\vert\vert A^*\vert=\vert A\vert^n\vert E\vert$,$\therefore\vert A^*\vert=\vert A\vert^{n-1}$。 + +又$AA^*=\vert A\vert E$,$\therefore A^*=\vert A\vert A^{-1}$,$(A^*)^{-1}=(\vert A\vert A^{-1})^{-1}=\dfrac{1}{\vert A\vert}A$。 + +$\because(A^*)^*=(A^*)^{-1}\vert A^*\vert E$,$\therefore=\dfrac{1}{\vert A\vert}A\vert A^*\vert=\dfrac{1}{\vert A\vert}A\vert A\vert^{n-1}=\vert A\vert^{n-2}A$。 + +\textbf{例题:}假设$A$为$n$阶方阵,求$(kA)^*$。 + +解:根据$AA^*=\vert A\vert E$,$\therefore (kA)(kA)^*=\vert kA\vert E$,推出$(kA)^*=\vert kA\vert(kA)^{-1}$。 + +$=k^n\vert A\vert\dfrac{1}{k}A^{-1}=k^{n-1}\vert A\vert A^{-1}=k^{n-1}A^*$。 + +\subsection{分块矩阵} + +在行列式的时候提到了分块行列式,分块行列式计算时要求对应的零行列式必须是行列数相等的,而对于分块矩阵而言则不要求,且不一定要零矩阵。 + +对于行列数较多的矩阵常使用\textbf{分块法},将大矩阵化为小矩阵。将矩阵用横纵线分为多个小矩阵,每个矩阵成为矩阵的\textbf{子块},以子块为元素的矩阵就是\textbf{分块矩阵}。 + +\subsubsection{分块矩阵计算} + +分块矩阵的计算法则与普通矩阵计算类似。 + +\textcolor{aqua}{\textbf{定理:}}若$AB$矩阵行列数相同,采用相同的分块法,则 \medskip + +$A=\left( + \begin{array}{ccc} + A_{11} & \cdots & A_{1r} \\ + \vdots & & \vdots \\ + A_{s1} & \cdots & A_{sr} + \end{array} +\right)\text{,}B=\left( + \begin{array}{ccc} + B_{11} & \cdots & B_{1r} \\ + \vdots & & \vdots \\ + B_{s1} & \cdots & B_{sr} + \end{array} +\right)$ + +$A+B=\left( + \begin{array}{ccc} + A_{11}+B_{11} & \cdots & A_{1r}+B_{1r} \\ + \vdots & & \vdots \\ + A_{s1}+B_{s1} & \cdots & A_{sr}+B_{sr} + \end{array} +\right)\text{。}$ + +\textcolor{aqua}{\textbf{定理:}}设$A=\left( + \begin{array}{ccc} + A_{11} & \cdots & A_{1r} \\ + \vdots & & \vdots \\ + A_{s1} & \cdots & A_{sr} + \end{array} +\right)$,$\lambda$为数,则$\lambda A=\left( + \begin{array}{ccc} + \lambda A_{11} & \cdots & \lambda A_{1r} \\ + \vdots & & \vdots \\ + \lambda A_{s1} & \cdots & \lambda A_{sr} + \end{array} +\right)$。\medskip + +\textcolor{aqua}{\textbf{定理:}}若$A_{m\times l}$,$B_{l\times n}$,采用相同的分块法,则 \medskip + +$A=\left( + \begin{array}{ccc} + A_{11} & \cdots & A_{1t} \\ + \vdots & & \vdots \\ + A_{s1} & \cdots & A_{st} + \end{array} +\right)\text{,}B=\left( + \begin{array}{ccc} + B_{11} & \cdots & B_{1t} \\ + \vdots & & \vdots \\ + B_{t1} & \cdots & B_{sr} + \end{array} +\right)$ + +$AB=\left( + \begin{array}{ccc} + C_{11} & \cdots & C_{1r} \\ + \vdots & & \vdots \\ + C_{s1} & \cdots & C_{sr} + \end{array} +\right)\text{,}C_{ij}=\sum\limits_{k=1}^tA_{ik}B_{kj}\text{。}$ + +\textcolor{aqua}{\textbf{定理:}}设$A=\left( + \begin{array}{ccc} + A_{11} & \cdots & A_{1r} \\ + \vdots & & \vdots \\ + A_{s1} & \cdots & A_{sr} + \end{array} +\right)$,则$A^T=\left( + \begin{array}{ccc} + A_{11}^T & \cdots & A_{s1}^T \\ + \vdots & & \vdots \\ + A_{1r}^T & \cdots & A_{sr}^T + \end{array} +\right)$。 \medskip + +\textcolor{aqua}{\textbf{定理:}}设$A$为$n$阶方阵,$A$的分块矩阵只有对角线上才有非零子块且都是方阵,其余子块都是零矩阵,即$A=\left( + \begin{array}{cccc} + A_1 & & & O \\ + & A_2 & \\ + & & \ddots & \\ + O & & & A_s + \end{array} +\right)$,称为\textbf{分块对角矩阵}。$\vert A\vert=\vert A_1\vert\vert A_2\vert\cdots\vert A_s\vert$。 + +若$\vert A_i\vert\neq0$,则$\vert A\vert\neq0$,且$A^{-1}=\left( + \begin{array}{cccc} + A_1^{-1} & & & O \\ + & A_2^{-1} & \\ + & & \ddots & \\ + O & & & A_s^{-1} + \end{array} +\right)$。 + +\subsubsection{按行按列分块} + +对于$m\times n$的矩阵$A$,其$n$列称为$A$的$n$个列向量,若第$j$列记为$a_j=\left( + \begin{array}{c} + a_{1j} \\ + a_{2j} \\ + \vdots \\ + a_{mj} + \end{array} +\right)$,则$A$可以按列分块为$A=(a_1,a_2,\cdots,a_n)$。\medskip + +其$m$行称为$A$的$m$个行向量,若第$i$行记为$a_i^T=(a_{i1},a_{i2},\cdots,a_{in})$,则$A$可以按行分块为$A=\left(\begin{array}{c} + a_1^T \\ + a_2^T \\ + \vdots \\ + a_{m}^T +\end{array}\right)$。 + +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:\medskip + +$AB=\left( + \begin{array}{c} + a_1^T \\ + a_2^T \\ + \vdots \\ + a_{m}^T + \end{array} +\right)(b_1,b_2,\cdots,b_n)$ + +$=\left( + \begin{array}{cccc} + a_1^Tb_1 & a_1^Tb_2 & \cdots & a_1^Tb_n \\ + a_2^Tb_1 & a_2^Tb_2 & \cdots & a_2^Tb_n \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{m}^Tb_1 & a_{m}^Tb_2 & \cdots & a_{m}^Tb_n + \end{array} +\right)=(c_{ij})_{m\times n}\text{。}$ + +其中:$c_{ij}=a_i^Tb_j=(a_{i1},a_{i2},\cdots,a_{is})\left(\begin{array}{c} + b_{1j} \\ + b_{2j} \\ + \vdots \\ + b_{sj} +\end{array}\right)=\sum\limits_{k=1}^s=a_{ik}b_{kj}\text{。}$ + +\textcolor{aqua}{\textbf{定理:}}$A=O$的充要条件是$A^TA=O$。 + +证明:$\because A=O$,$\therefore A^T=O$,$A^TA=O$。 + +设$A=(a_{ij})_{m\times n}$,将$A$按列分块为$A=(a_1,a_2,\cdots,a_n)$,则 \medskip + +$A^TA=\left( + \begin{array}{c} + a_1^T \\ + a_2^T \\ + \vdots \\ + a_{m}^T + \end{array} +\right)(a_1,a_2,\cdots,a_n)=\left( + \begin{array}{cccc} + a_1^Ta_1 & a_1^Ta_2 & \cdots & a_1^Ta_n \\ + a_2^Ta_1 & a_2^Ta_2 & \cdots & a_2^Ta_n \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{m}^Ta_1 & a_{m}^Ta_2 & \cdots & a_{m}^Ta_n + \end{array} +\right)\text{。}$\medskip + +所以$A^TA$的元为$a^T_ia_j$,又$\because A^TA=O$,$\therefore a^T_ia_j=0$($i,j=1,2,\cdots n$)。 + +$\therefore a^T_ja_j=0$($j=1,2,\cdots n$),对角线元素全部为0。\medskip + +且$a^T_ja_j=\left( + \begin{array}{cccc} + a_1^Ta_1 & & & \\ + & a_2^Ta_2 & & \\ + & & \ddots & \\ + & & & a_{m}^Ta_n + \end{array} +\right)=(a_{1j},a_{2j},\cdots,a_{mj})\left(\begin{array}{c} + a_{1j} \\ + a_{2j} \\ + \vdots \\ + a_{mj} +\end{array}\right)$ \medskip + +$=a_{1j}^2+a_{2j}^2+\cdots+a_{mj}^2=0$,所以$a_{1j}=a_{2j}=\cdots+a_{mj}=0$。 + +$\therefore A=O$。 + +\section{线性方程组} + +矩阵是根据线性方程组得到。 + +\subsection{线性方程组与矩阵} + +\begin{multicols}{2} + + $\begin{cases} + a_{11}x_1+\cdots+a_{1n}x_n=0 \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n=0 + \end{cases}$ \medskip + + $n$元齐次线性方程组。 + + $\begin{cases} + a_{11}x_1+\cdots+a_{1n}x_n=b_1 \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n=b_n + \end{cases}$ \medskip + + $n$元非齐次线性方程组。 + +\end{multicols} + +对于齐次方程,$x_1=\cdots=x_n=0$一定是其解,称为其\textbf{零解},若有一组不全为零的解,则称为其\textbf{非零解}。其一定有零解,但是不一定有非零解。 + +对于非齐次方程,只有$b_1\cdots b_n$不全为零才是。\medskip + +令\textbf{系数矩阵}$A_{m\times n}=\left( + \begin{array}{ccc} + a_{11} & \cdots & a_{1n} \\ + \cdots \\ + a_{m1} & \cdots & a_{mn} + \end{array} +\right)$,\textbf{未知数矩阵}$x_{n\times 1}=\left( + \begin{array}{c} + x_1 \\ + \cdots \\ + x_n + \end{array} +\right)$,\textbf{常数项矩阵}$b_{m\times 1}=\left( + \begin{array}{c} + b_1 \\ + \cdots \\ + b_m + \end{array} +\right)$,\textbf{增广矩阵}$B_{m\times(n+1)}=\left( + \begin{array}{c:c} + \begin{matrix} + a_{11} & \cdots & a_{1n}\\ + \cdots \\ + a_{m1} & \cdots & a_{mn} + \end{matrix}& + \begin{matrix} + b_1\\ + \\ + b_n + \end{matrix} + \end{array} +\right)$。 + +所以$AX=\left( + \begin{array}{c} + a_11x_1+\cdots+a_{1n}x_n \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n + \end{array} +\right)$。 + +从而$AX=b$等价于$\begin{cases} + a_{11}x_1+\cdots+a_{1n}x_n=b_1 \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n=b_n +\end{cases}$,当$b=O$就是齐次线性方程。 + +从而矩阵可以简单表示线性方程。 + +\subsection{矩阵乘法与线性变换} + +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。 + +$\begin{cases} + y_1=a_{11}x_1+a_{12}x_2+\cdots+a_{1s}x_s \\ + \cdots \\ + y_m=a_{m1}x_1+a_{m2}x_2+\cdots+a_{ms}x_s +\end{cases}\begin{cases} + x_1=b_{11}t_1+b_{12}t_2+\cdots+b_{1n}t_n \\ + \cdots \\ + x_s=b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n +\end{cases}$\medskip + +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:\medskip + +$\begin{cases} + y_1=a_{11}(b_{11}t_1+\cdots+b_{1n}t_n)+\cdots+a_{1s}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) \\ + \cdots \\ + y_m=a_{m1}(b_{11}t_1+\cdots+b_{1n}t_n)+\cdots+a_{ms}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) +\end{cases}$ + +$=\begin{cases} + y_1=(a_{11}b_{11}+\cdots+a_{1s}b_{s1})t_1+\cdots+(a_{11}b_{1n}+\cdots+a_{1s}b_{sn})t_n \\ + \cdots \\ + y_m=(a_{m1}b_{11}+\cdots+a_{ms}b_{s1})t_1+\cdots+(a_{m1}b_{1n}+\cdots+a_{ms}b_{sn})t_m +\end{cases}$ \medskip + +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:\medskip + + +$\left(\begin{array}{ccc} + a_{11} & \cdots & a_{1s} \\ + \vdots & \ddots & \vdots \\ + a_{m1} & \cdots & a_{ms} +\end{array}\right)_{m\times s}\left(\begin{array}{ccc} + b_{11} & \cdots & a_{1n} \\ + \vdots & \ddots & \vdots \\ + b_{s1} & \cdots & b_{sn} +\end{array}\right)_{s\times n}$ + +$=\left(\begin{array}{ccc} + a_{11}b_{11}+\cdots+a_{1s}b_{s1} & \cdots & a_{11}b_{1n}+\cdots+a_{1s}b_{sn} \\ + \vdots & \ddots & \vdots \\ + a_{m1}b_{11}+\cdots+a_{ms}b_{s1} & \cdots & a_{m1}b_{1n}+\cdots+a_{ms}b_{sn} +\end{array}\right)_{m\times n}\text{。}$ + +\subsection{线性方程组的解} + +对于一元一次线性方程:$ax=b$: + +\begin{itemize} + \item 当$a\neq 0$时,可以解得$x=\dfrac{b}{a}$。 + \item 当$a=0$时,若$b\neq 0$时,无解,若$b=0$时,无数解。 +\end{itemize} + +当推广到多元一次线性方程组:$Ax=b$,如何求出$x$这一系列的$x$的解? + +从数学逻辑上看,已知多元一次方程,有$m$个约束方程,有$n$个未知数,假定$m\leqslant n$。 + +当$mJZZ7}8GIJ0y5!o48!t(Mmid$Ga105N~tqq-lqCjIi 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+\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\usepackage{amssymb} +% 因为所以 +\usepackage{amsmath} +% 数学公式 +\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} +% 超链接 +\author{Didnelpsun} +\title{向量} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} + +线性代数的主要研究对象就是向量,行列式与矩阵都是由向量组成的向量组。 + +\section{向量与向量组} + +\subsection{向量的定义与运算} + +$n$维向量\textcolor{violet}{\textbf{定义:}}$n$个数构成的一个有序数组$[a_1,a_2,\cdots,a_n]$称为一个$n$维向量,记为$\alpha=[a_1,a_2,\cdots,a_n]$,并称$\alpha$为$n$维行向量,$\alpha^T$为$n$维列向量,$a_i$为向量$\alpha$的$i$个分量。 + +若$\alpha$与$\beta$都是$n$维向量,且对应元素相等,则$\alpha=\beta$。 + +$\alpha+\beta=[a_1+b_1,a_2+b_2,\cdots,a_n+b_n]$。 + +$k\alpha=[ka_1,ka_2,\cdots,ka_3]$。 + +\subsection{向量组的线性概念} + +线性组合$\textcolor{violet}{\textbf{定义:}}$$m$个$n$维向量$\alpha_1,\alpha_2,\cdots,\alpha_m$以及$m$个数$k_1,k_2,\cdots,k_m$,则向量$k_1\alpha_1+k_2\alpha_2+\cdots+k_m\alpha_m$就是向量组$a_1,a_2,\cdots,a_m$的线性组合。 + +线性表出$\textcolor{violet}{\textbf{定义:}}$若向量$\beta$能表示成向量组$\alpha_1,\alpha_2,\cdots,a_m$的线性组合,则存在$m$个数$k_1,k_2,\cdots,k_m$,使得$\beta=k_1\alpha_1+k_2\alpha_2+\cdots+k_m\alpha_m$,则成向量$\beta$能被向量组$a_1,a_2,\cdots,a_m$线性表出。否则不能被线性表出。 + +线性相关$\textcolor{violet}{\textbf{定义:}}$对$m$个$n$维向量$a_1,a_2,\cdots,a_m$,存在一组不全为0的数$k_1,k_2,\cdots,k_m$,使得$k_1\alpha_1+k_2\alpha_2+\cdots+k_m\alpha_m=0$,则称$a_1,a_2,\cdots,a_m$线性相关。 + +含有零向量或成比例向量的向量组必然线性相关。 + +线性无关$\textcolor{violet}{\textbf{定义:}}$对$m$个$n$维向量$a_1,a_2,\cdots,a_m$,不存在一组不全为0的数$k_1,k_2,\cdots,k_m$,使得$k_1\alpha_1+k_2\alpha_2+\cdots+k_m\alpha_m=0$,即仅当$k_1=k_2=\cdots=k_m=0$才成立,则称$a_1,a_2,\cdots,a_m$线性无关。 + +两个非零向量,不成比例向量的向量必然线性无关。 + +\end{document}