From 8c2015cbbd39c27bf851721a67ca96f6e2293e13 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Tue, 6 Jul 2021 23:25:17 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=BA=BF=E6=80=A7=E6=96=B9?= =?UTF-8?q?=E7=A8=8B=E7=BB=84?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- linear-algebra/exercise/3-vector/vector.pdf | Bin 25359 -> 68481 bytes linear-algebra/exercise/3-vector/vector.tex | 143 +++++++++- linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 134648 -> 120267 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 208 ++------------ linear-algebra/knowledge/3-vector/vector.pdf | Bin 38126 -> 75056 bytes linear-algebra/knowledge/3-vector/vector.tex | 80 +++++- .../linear-equations-system.pdf | Bin 0 -> 68980 bytes .../linear-equations-system.tex | 264 ++++++++++++++++++ 8 files changed, 501 insertions(+), 194 deletions(-) create mode 100644 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zEp!5!#&^eD%ngI!yK%pl7k+R|BKh3UV|C8YWh_s=JH2xQ;tQhui>?p zb8@6M`Y*1}zY$_nl`b-C^7)H2yKPB~N6&N>Am1{TV7jQlHrnIoMVENah1bMXn#0KV z55DFVIy=_^BM07aav`w;*=V}A^#{;xQfrL{FWKbT$H+dhjkHajN8|ZLOsDjp*qx#Y zpqJd=_aQD_Z+O+!ct0p*09K@95S!2oEUWH(c3pi=wy>9!fgb96^zLZquq2R;KPRLTWbd2&Kd!m@bQ(mqOoc6% zkts_`QSGzPGQPC|J-^vIF9?JN_)$2X)RM>N>*Br_lT@}_8=-Q0RgR_}}Z5%9#g5vMH6#Sq| zCIbO3F7asWSCm7(L`?IjN>p4*7rd)9)DJs$0AsrAhOjwXGR2PVc{++hW4ZzHp(E60 zH~RPzmc_Di9LL_OJoaWQ?}gW|0^xl+WWav1M@L(zwhu>AcIkkk)fg#PC+@zvx1Q|9 zq@Or$**lP*piqN6H!M$He<~~{aPif_nL3~5k%zI}1}|SB$m+~83Zn_2&+foH^H{(h zwV(Qy>iETnPHf+U1jA})+6pL zg(uG!;zHg>a}eTiylyZaL#$&#i@Szy>-NNMMve+$frk0}7mlrbbTFF~$wE@_MK?2^ z+WI6rPxcHbpeFyM#V(K(rn9-wa83%-x*csulxvn$时,有$n$个方程$s$个未知数,所以必然存在自由变量,从而必然线性相关性。 + +当$s=n$时,$\vert\alpha_1 \alpha_2 \cdots \alpha_n\vert=\left|\begin{array}{cccc} + 1 & 1 & \cdots & 1 \\ + a_1 & a_2 & \cdots & a_n \\ + \vdots & \vdots & \ddots & \vdots \\ + a_1^{n-1} & a_2^{n-1} & \cdots & a_n^{n-1} +\end{array}\right|=\prod\limits_{1\leqslant j\leqslant i\leqslant n}(a_i-a_j)\neq0$。所以线性无关。 + +当$sn$时线性相关,$s\leqslant n$时线性无关。 + +\section{极大线性无关组与向量组秩} + +极大线性无关组一般与向量组秩在一起使用。一般解出极大线性无关组与秩,还要用极大线性无关组表示出其余的向量,基本步骤: + +\begin{enumerate} + \item 将向量组拼接为矩阵$A$,对$A$进行初等行变换,化为最简行阶梯形矩阵,确定矩阵秩$r(A)$。 + \item 在最简行阶梯矩阵中按列找出一个秩为$r(A)$的子矩阵,即在每个台阶上找一列列向量,找$r(A)$列构成一个新矩阵,其就是一个极大线性无关组。 + \item 将其余向量依次与极大线性无关组进行对比解出表示方法。 +\end{enumerate} + +\textcolor{orange}{注意:}求向量组的秩可以进行初等变换,包括行变换和列变换。但是求极大线性无关组时最好只使用行变换,因为列变换会改变方程的解。从而解方程组只能做行变换。 + +\section{等价向量组} + +$r(A)=r(B)=r(A|B)$,所以需要计算三个向量组构成的矩阵的秩就可以了。 + +\textbf{例题:}设向量组$\alpha$:$\alpha_1=[1,0,2]^T$,$\alpha_2=[0,1,1]^T$,$\alpha_3=[2,-1,a+4]^T$,向量组$\beta$:$\beta_1=[1,2,4]^T$,$\beta_2=[1,-1,a+2]^T$,$\beta_3=[3,3,10]^T$。 + +矩阵$A=\left(\begin{array}{ccc} + 1 & 0 & 2 \\ + 0 & 1 & -1 \\ + 2 & 1 & a+4 +\end{array}\right)$,$B=\left(\begin{array}{ccc} + 1 & 1 & 3 \\ + 2 & -1 & 3 \\ + 4 & a+2 & 10 +\end{array}\right)$。\medskip + +(1)$AB$是否等价。 + +(2)向量组$AB$是否等价。 + +(1)解:化简$A=\left(\begin{array}{ccc} + 1 & 0 & 2 \\ + 0 & 1 & -1 \\ + 0 & 0 & a+1 +\end{array}\right)$,$B=\left(\begin{array}{ccc} + 1 & 3 & 1 \\ + 0 & 1 & 1 \\ + 0 & 0 & a +\end{array}\right)$ + +若$a\neq-1$,则$r(A)=3$,且$a\neq0$,则$r(B)=3$,此时$AB$等价。 + +若$a=-1$,则$r(A)=2$,$r(B)=3$,$AB$不等价。 + +若$a=0$,则$r(B)=2$,$r(A)=2$,$AB$不等价。 + +(2)解:因为向量组$\alpha$拼接在一起就是$A$,$\beta$拼接在一起就是$B$,所以$r(\alpha)=r(A)$,$r(\beta)=r(B)$,$r(\alpha|\beta)=r(A|B)$。 + +将$AB$拼在一起做行变换,得到$(A|B)=\left(\begin{array}{c:c} + \begin{matrix} + 1 & 0 & 2 \\ + 0 & 1 & -1 \\ + 0 & 0 & a+1 + \end{matrix}& + \begin{matrix} + 1 & 1 & 3 \\ + 2 & -1 & 3 \\ + 0 & a+1 & 1 + \end{matrix} +\end{array}\right)$。\medskip + +若$a\neq-1\neq0$,则$r(A)=r(B)=r(A|B)$。向量组等价。 + +若$a=-1$或$a=0$,则$r(A)\neq r(B)$,所以不等价。 + +\section{向量空间} + +\textbf{例题:}设$R^3$中有两个基$A$:$\alpha_1=[1,1,0]^T$,$\alpha_2=[0,1,1]^T$,$\alpha_3=[1,0,1]^T$,基$B$:$\beta_1=[1,0,0]^T$,$\beta_2=[1,1,0]^T$,$\beta_3=[1,1,1]^T$。 + +(1)求基$B$到基$A$的过渡矩阵。 + +(2)已知$\xi$在基$B$下的坐标为$[1,0,2]^T$,求$\xi$在基$A$下的坐标。 + +(1)解:过渡矩阵为$A=BC$,即$B^{-1}A=C$。 + +(2)解:令在基$A$下的坐标为$(x_1,x_2,x_3)^T$。 + +$\therefore\xi=A(x_1,x_2,x_3)^T=B(1,0,2)^T$,$(x_1,x_2,x_3)^T=A^{-1}B(1,0,2)^T$。 \end{document} diff --git a/linear-algebra/knowledge/2-matrix/matrix.pdf b/linear-algebra/knowledge/2-matrix/matrix.pdf index a926e1bce8e2fae3105a6157cdffb19cdaee2584..987d69418a8db5b39e5a7d733e37ac145266b049 100644 GIT binary patch delta 99302 zcmZsiV{k6cvhHK8m@Bqz+qP}nd}G_TZF9x8ZQIF8a{qgubF0q2=YE-*s;T)hJ=4GL zr>EbtpeCyz}XLas~cX2-y@n%l1Zj~hRHwK_cuVc>KXy`FG_DYsXNc)i;R z++UVk(M^r!P;l!4%Q^C>0YhY)+XA95e3Z9et2Q?NCU%E!`8~X6p%!=aPi?Z$-iR(3 zHp`xr&|`qGFPXlI7bPAqaa;D`$*C1^`t5s_Q{bSrDgV+sehh&r>gUVJb2>NC5Hz3L=%i1Z<=hkYoQV z6JoO_LX_Z?TFNB>Ly)@q#^1UUKMI_X=EZ%7R%(DkJaVk-vz1|#FAhNlkhq<5z@eZ+ z^1_V&Af?DY2z*-QJ^?fuH-;ve@u9T$uipUVRGo`wSPd3`Eo6m$p)jeW(eruzjx|gBMD==sBIQlIMQ~{K!BArjPy6nJZy14Wf_FN#oPpkx za;o%De!a;F&~oD?gkiro2;+VOrxSUc3=#p}SO=Utjw<)erEAdf;^Q9<$6A!CJXT|m zSD;N-wr2diE2x^Nk~+L3ELYJ?)dY&?H31w%%3Wg4hC*stS$Z!OU5#0gugT#sOu>YS 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a/linear-algebra/knowledge/2-matrix/matrix.tex b/linear-algebra/knowledge/2-matrix/matrix.tex index e9f170b..812b75d 100644 --- a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -28,6 +28,7 @@ \usepackage{multicol} % 分栏 \usepackage{arydshln} +% 增广矩阵长虚线 \setlength{\dashlinegap}{1pt} \setlength{\dashlinedash}{1pt} % 阶梯矩阵的虚线 @@ -510,195 +511,6 @@ $=a_{1j}^2+a_{2j}^2+\cdots+a_{mj}^2=0$,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$mD)Z1U~hy}NmnPN$Rpuj+KwIaTM^ zr1BzSw2XADP^3q9*Kbfv>;w!1c7~QvJUsMb7S_%tj`U*I2F@lTCPsF~CiK!Kwr0-e z1ZX7^Wx+IA&rbCa;t6!k=-GSB7J84VS(feT_Q&F{_X3ua~c}TCYFM zoMI4vmuJ_!|~j4E_!|X#KaF6czkw!>)&_c{Zso6zoOT}*z`aWP6xe@a0}|T 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zx62ObiF60&HU9>ouT<0gMyZNS?p>7-ZHXq?6g#IDAz???DfV8ONOO%yL0jV`FOpz1 znrexA;DifRz(qoH9@lgAW@h!|9V%>-d))K(W-ZG79Bx#e?QFqM#g`Sh+!L;0bTD12 z#0JnyU&;@~pSz+Tea6_WKiHEkW3(>eOfPj3C2t|tFHBawKW%!SSh>C`_=BUv9w2(> l<+r#29WqV)-(&!KQwU^=AKA&7M^zoJsmi0Epl@ct^Di!~VXgoG diff --git a/linear-algebra/knowledge/3-vector/vector.tex b/linear-algebra/knowledge/3-vector/vector.tex index 21280ff..e00b44a 100644 --- a/linear-algebra/knowledge/3-vector/vector.tex +++ b/linear-algebra/knowledge/3-vector/vector.tex @@ -25,6 +25,8 @@ % 数学公式 \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} % 超链接 +\usepackage{pifont} +% 圆圈序号 \author{Didnelpsun} \title{向量} \date{} @@ -54,16 +56,86 @@ $k\alpha=[ka_1,ka_2,\cdots,ka_3]$。 \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$线性相关。 +线性相关\textcolor{violet}{\textbf{定义:}}对$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$线性无关。 +线性无关\textcolor{violet}{\textbf{定义:}}对$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$线性无关。 两个非零向量,不成比例向量的向量必然线性无关。 +\subsection{线性相关性} + +\begin{enumerate} + \item 向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$($n\geqslant2$)线性相关的充要条件是向量组中至少有一个向量可由其他$n-1$个向量线性表出。若$\alpha_1,\alpha_2,\cdots,\alpha_n$线性无关的充要条件是向量组的任何一个向量都不能被其他$n-1$个向量线性表出。 + \item 向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$线性无关,而$\beta,\alpha_1,\alpha_2,\cdots,\alpha_n$线性相关,则$\beta$可由$\alpha_1,\alpha_2,\cdots,\alpha_n$线性表示,且表示方法唯一。 + \item 向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$可由向量组$\beta_1,\beta_2,\cdots,\beta_s$线性表示,且$n>s$,则$\alpha_1,\alpha_2,\cdots,\alpha_n$线性相关。(以少表多,多的相关)若向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$可由向量组$\beta_1,\beta_2,\cdots,\beta_s$线性表示,$\alpha_1,\alpha_2,\cdots,\alpha_n$线性无关,则$n\leqslant s$。 + \item 设$m$个$n$维向量$\alpha_1,\alpha_2,\cdots,\alpha_m$,其中$\alpha_1=[a_{11},a_{12},\cdots,a_{m1}]^T$,$\cdots$,$\alpha_m=[a_{1m},a_{2m},\cdots,a_{mm}]^T$,则向量组$\alpha_1,\alpha_2,\cdots,\alpha_m$线性相关的充要条件是齐次线性方程$Ax=0$有非零解,其中$A=[\alpha_1,\alpha_2,\cdots,\alpha_m]$,$x=[x_1,x_2,\cdots,x_m]^T$。$m$个$n$维向量$\alpha_1,\alpha_2,\cdots,\alpha_m$线性无关的充要条件是齐次线性方程$Ax=0$只有零解。 + \item 向量$\beta$可由向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$表出,则向量组$\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n=[\alpha_1,\alpha_2,\cdots,\alpha_n][x_1,x_2,\cdots,x_n]^T=\beta$有解,即$r([\alpha_1,\alpha_2,\cdots,\alpha_n])=r([\alpha_1,\alpha_2,\cdots,\alpha_n,\beta])$。否则则不能表出,则方程无解,$r([\alpha_1,\alpha_2,\cdots,\alpha_n])+1=r([\alpha_1,\alpha_2,\cdots,\alpha_n,\beta])$ + \item 向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$存在一部分向量线性相关,则整个向量组线性相关。若$\alpha_1,\alpha_2,\cdots,\alpha_n$线性无关,则任意一部分向量组线性无关。 + \item 设$m$个$n$维向量$\alpha_1,\alpha_2,\cdots,\alpha_m$线性无关,则把这些向量中每个各任意添加$s$个分量所得到的新向量组($n+s$维)$\alpha_1^*,\alpha_2^*,\cdots,\alpha_m^*$也是线性无关的;如果$\alpha_1,\alpha_2,\cdots,\alpha_m$线性相关,则每个各去掉相同的若干分量得到的新向量组也线性相关。(原来无关延长无关,原来相关缩短相关) +\end{enumerate} + +\section{极大线性无关组} + +\subsection{概念} + +极大线性无关组\textcolor{violet}{\textbf{定义:}}在向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$中,若存在部分$a_i,a_j,\cdots,a_k$满足:\ding{172}$a_i,a_j,\cdots,a_k$线性无关。\ding{173}向量组中任一向量$a_s$($i=1,2,\cdots,n$)均可由$a_i,a_j,\cdots,a_k$线性表出。则称向量组$a_i,a_j,\cdots,a_k$为原向量组的极大线性无关组。 + +不包含无用约束方程的最简方程组的系数矩阵就是极大线性无关组。 + +向量组的极大线性无关组一般不唯一,只由一个零向量组成的向量组不存在极大线性无关组,一个线性无关向量组的极大线性无关组就是其本身。 + +\section{向量组秩} + +向量组构成矩阵的秩等于行向量组的秩等于列向量组的秩。 + +若$A$通过初等行变换为$B$,则$AB$的行向量组是等价向量组,任何对应的部分列向量组都具有同样的线性相关性。 + +若向量组$B$均可由$A$线性表出,则$r(B)\leqslant r(A)$。 + +\section{等价向量组} + +任何一个组都可以由其极大线性无关组来代表。 + +\subsection{定义} + +设两个向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$和$\beta_1,\beta_2,\cdots,\beta_m$,若这两个向量组可以互相线性表出,则称其为等价向量组,记为$\alpha\cong\beta$。 + +具有的性质: + +\begin{enumerate} + \item $A\cong A$(反身性)。 + \item $A\cong B$,则$B\cong A$(对称性)。 + \item $A\cong B$,$B\cong C$,则$A\cong C$(传递性)。 +\end{enumerate} + +向量组和其极大线性无关组是等价向量组。 + +\subsection{判定} + +若$r(A)=r(B)=r(A|B)$,则向量组等价。 + +\subsection{与等价矩阵区别} + +对于矩阵而言,若$A\cong B$,则$AB$同型且$r(A)=r(B)$。 + +对于向量组而言,若$A\cong B$,则$AB$同维(行数相同)且$r(A)=r(B)=r(A|B)$。 + +\section{向量空间} + +\subsection{基本概念} + +若$\xi_1,\xi_2,\cdots,\xi_n$是$n$维向量空间$R^n$中的线性无关的有序向量组,则任意向量$\alpha\in R^n$均可由$\xi_1,\xi_2,\cdots,\xi_n$线性表出,记为$\alpha=a_1\xi_1+a_2\xi_2+\cdots+a_n\xi_n$,类似一个极大线性无关组,则称有序向量组$\xi_1,\xi_2,\cdots,\xi_n$为$R^n$的一个\textbf{基},基向量的个数$n$为向量空间的\textbf{维数},而$[a_1,a_2,\cdots,a_n]([a_1,a_2,\cdots,a_n]^T)$为向量$\alpha$在基$\xi_1,\xi_2,\cdots,\xi_n$下的\textbf{坐标},或称为$\alpha$的坐标行列向量。 + +\subsection{基变换与坐标变换} + +若$\eta_1,\eta_2,\cdots,\eta_n$和$\xi_1,\xi_2,\cdots,\xi_n$是$R^n$中两个基,且有关系:$[\eta_1,\eta_2,\cdots,\eta_n]=[\xi_1,\xi_2,\cdots,\xi_n]C_{n\times n}$,则这个式子称为基$\xi_1,\xi_2,\cdots,\xi_n$到基$\eta_1,\eta_2,\cdots,\eta_n$的\textbf{基变换公式},矩阵$C$就是基$\xi_1,\xi_2,\cdots,\xi_n$到基$\eta_1,\eta_2,\cdots,\eta_n$的\textbf{过渡矩阵},$C$可逆,$C$的第$i$列就是$\eta_i$在基$\xi_1,\xi_2,\cdots,\xi_n$下的坐标列向量。 + +$\alpha$在基$\xi_1,\xi_2,\cdots,\xi_n$和基$\eta_1,\eta_2,\cdots,\eta_n$下坐标分别为$x=[x_1,x_2,\cdots,x_n]^T$,$y=[y_1,y_2,\cdots,y_n]^T$,即$\alpha=[\xi_1,\xi_2,\cdots,\xi_n]x=[\eta_1,\eta_2,\cdots,\eta_n]y$。又$C$是基$\xi_1,\xi_2,\cdots,\xi_n$到基$\eta_1,\eta_2,\cdots,\eta_n$的过渡矩阵,则$[\xi_1,\xi_2,\cdots,\xi_n]=[\eta_1,\eta_2,\cdots,\eta_n]C$,则$\alpha=[\xi_1,\xi_2,\cdots,\xi_n]x=[\eta_1,\eta_2,\cdots,\eta_n]y=[\xi_1,\xi_2,\cdots,\xi_n]Cy$,从而$x=Cy$或$y=C^{-1}x$,这个就是\textbf{坐标变换公式}。 + \end{document} diff --git a/linear-algebra/knowledge/4-linear-equations-system/linear-equations-system.pdf 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+\usepackage{multicol} +% 分栏 +\usepackage{arydshln} +% 增广矩阵长虚线 +\author{Didnelpsun} +\title{线性方程组} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{基本概念} + +矩阵是根据线性方程组得到。线性方程组和向量组本质上是一致的。 + +\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} + +$m$是方程个数,即方程组行数,$n$是方程未知数个数,即类似方程组的列数。 + +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$m