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A^{k-1}\alpha\neq0$,$\therefore\lambda_2=0$,消去$\lambda_2$:$\ 当谈到多个向量是否线性相关时可以将向量组组成矩阵,判断其秩。满秩就是线性无关,降秩就是线性相关。 -\section{线性表出} - -\subsection{矩阵秩} - 当谈到一个向量是否能被其他向量线性表出时,要将这些向量全部组成一起,判断能否被其他向量表出的向量放在最右边,然后判断增广矩阵的秩。 \begin{enumerate} @@ -153,6 +149,8 @@ $=[\alpha_1,\alpha_2,\alpha_3|\beta]=\left[\begin{array}{cccc} $\therefore a=3$。 +\section{线性表出} + \subsection{极大线性无关组} 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: @@ -179,6 +177,28 @@ $\therefore a=3$。 对其行变换,解得$a=5$。 +\subsection{向量线性表示} + +即要求$\beta$关于$\alpha_i$的线性表出表达式。 + +基本方法是设$\beta=a\alpha_1+b\alpha_2+\cdots$,然后每行代入求出,不过也可以使用矩阵变换法。 + +可以同时求多个$\beta$的表示方式,设$\alpha_i$为长度为$h$的列向量,一共有$n$个,组成$A=(\alpha_1,\alpha_2,\cdots,\alpha_n)$,设$\beta_i$为长度为$h$的列向量,一共有$n$个,组成$B=(\beta_1,\beta_2,\cdots,\beta_n)$。 + +$[A|B]$通过线性变换得到$[E|C]$,则$B=CA$。 + +\textbf{例题:}用$\alpha_1=(1,1,1)^T$、$\alpha_2=(1,2,4)^T$、$\alpha_3=(1,3,9)^T$表示$\beta=(1,1,3)^T$。 + +解:组成矩阵$\left[\begin{array}{cccc} + 1 & 1 & 1 & 1 \\ + 1 & 2 & 3 & 1 \\ + 1 & 4 & 9 & 3 +\end{array}\right]=\left[\begin{array}{cccc} + 1 & 0 & 0 & 2 \\ + 0 & 1 & 0 & -2 \\ + 0 & 0 & 1 & 1 +\end{array}\right]$,所以$\beta=2\alpha_1-2\alpha_2+\alpha_3$。 + \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。 diff --git a/linear-algebra/knowledge/4-linear-equations-system/linear-equations-system.pdf b/linear-algebra/knowledge/4-linear-equations-system/linear-equations-system.pdf index a76a056710d99396691089810acaac9571fd0e4b..90d58e4c2cb10999996a0c8197326e6d0e207d59 100644 GIT binary patch delta 24226 zcmZU(Q;;r97p&Q~ZQHhOW4CSF{qDAHYqxFNwr$&iUVuwJNi6aS3vF z39`<~4uqR2iI4*WaMlJ|l;M4k-U_(`j{@3`ou0J{$*;`~h7kXjn3qSgtRjn3vCO}C zDGxM=Wf>-_k?{QWM_lu_q7LL6kybIqV?Ov#bMILn;>Y9b<1Aw1ZrE*U+C?1T89ma- zNv-<{9U@3%8F*gIrZVlOloUu32p35A@#}GN*EJ%C(TJHDK-12arm!Y^iDQg)YRo=2 z;ex6MwJf{}Hcs2-aQD6!Czd-yd*BOR4F4YR6&9xmo+(AK&<0$V)glIwBW%VO|E@+h zI5O@U+mg^kdp=pig^>%PF)0xHz% zkKtGSSbFrI6)zt6$EM%t20a+fI0%b<|F@g-KC08j>Lwx6z(-!~F_lq{U-$@!a-MLl 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