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z<;nk22p6fAj!=hlDo~G@e+p4Dh({=4vy9q+xF#aG`b1J{J+|KVDc5!&E`2pUN6T9{yp2|h3S2YGlmmu%Oke0{9K-j|wA z@^b@ttS>pm6y6oY0*o>8X*Du(@>}v87~`pkvCnlWEfd=D*Z!ljDo#?hcP)t&YSZ-$ zy1kk1cBxvI?nUxe;+CB)mr9JwagSl9ifdKQ71G;Z=LHD5K7QYT9s?iEB?HG$H2DAX cu{b#!I6AvKnwTQ6adUF9B2Z9>%YQ}qKS9H#y#N3J diff --git a/linear-algebra/exercise/4-linear-equations-system/linear-equations-system.tex b/linear-algebra/exercise/4-linear-equations-system/linear-equations-system.tex index 5efc0c4..30a85c3 100644 --- a/linear-algebra/exercise/4-linear-equations-system/linear-equations-system.tex +++ b/linear-algebra/exercise/4-linear-equations-system/linear-equations-system.tex @@ -60,7 +60,7 @@ 代入$Ax=b$:$A(3\alpha_1-2\alpha_2)=3A\alpha_1-2A\alpha_2=3b-2b=b$,所以成立。 -\subsection{线性表出} +% \subsection{线性表出} \section{反求参数} @@ -91,6 +91,36 @@ $\vert A\vert=\left\vert\begin{array}{ccc} 解得$a=-5$或$a=-6$。 +\section{抽象线性方程} + +\textbf{例题:}设$A$为三阶方阵,$A=(a_{ij})_{3\times3}$,且$a_{ij}=A_{ij}$($i,j=1,2,3$),其中$A_{ij}$为$a_{ij}$的代数余子式,$a_{33}\neq0$,$b=(a_{13},a_{23},a_{33})^T$,求非齐次线性方程组$Ax=b$的解。 + +解:由于是抽象线性方程,所以必须要充分利用方程和矩阵的性质。题目中给出的主要是代数余子式,由行列式的一行或一列的元素乘上对应的代数余子式可得行列式值的性质: + +$b$为第三列元素,乘上对应的代数余子式得行列式值:$b(A_{13},A_{23},A_{33})=a_{13}A_{13}+a_{23}A_{23}=a_{13}^2+a_{23}^2+a_{33}^2\geqslant0$。 + +又$a_{33}\neq0$,$\therefore b(A_{13},A_{23},A_{33})=\vert A\vert>0$,$r(A)=3$,$Ax=b$解唯一$x=A^{-1}b$。 + +根据逆矩阵公式$x=A^{-1}b=\dfrac{A^*b}{\vert A\vert}$,且代数余子式乘上非对应元素值都为0。 + +$=\dfrac{1}{\vert A\vert}\left[\begin{array}{ccc} + A_{11} & A_{21} & A_{31} \\ + A_{12} & A_{22} & A_{32} \\ + A_{13} & A_{23} & A_{33} +\end{array}\right]\left[\begin{array}{c} + a_{13} \\ + a_{23} \\ + a_{33} +\end{array}\right]=\dfrac{1}{\vert A\vert}\left[\begin{array}{c} + 0 \\ + 0 \\ + \vert A\vert +\end{array}\right]=\left[\begin{array}{c} + 0 \\ + 0 \\ + 1 +\end{array}\right]$。 + \section{公共解} \end{document} \ No newline at end of file diff --git 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\begin{turn}{80}$\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|$。 + \subsubsection{分块行列式} $\left|\begin{array}{cc} diff --git a/linear-algebra/knowledge/4-linear-equations-system/linear-equations-system.pdf b/linear-algebra/knowledge/4-linear-equations-system/linear-equations-system.pdf index 22acceb26709c607eb437ad0ff2c6697d4a82c97..a76a056710d99396691089810acaac9571fd0e4b 100644 GIT binary patch delta 92849 zcmZ6yV~}RivMt(W+qP}nw!3WG_{z4;F59+k+v>9YdY^k=#M$>>#{4s5WvAl9V)CNcmEDK7zD?FlBv4Jx39 zp_+{RN*EH&WP~}J>eu4Ymv2V~ocpw7Opo{?)}dFPHvq&MQlrr@6274wjALbGB9x_0 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b/linear-algebra/knowledge/4-linear-equations-system/linear-equations-system.tex @@ -505,12 +505,18 @@ $ABD$选项增广矩阵的秩都为3,所以不能表示,而只有$C$的为2 \section{公共解} +\subsection{待定系数法} + \begin{enumerate} \item 求两个方程组解的交集部分。可以联立两个方程求解。 \item 求出$A_{m\times n}x=0$的通解$k_1\xi_1+k_2\xi_2+\cdots+k_s\xi_s$,这些$k$本来是独立的,然后代入$B_{m\times n}x=0$,求出$k_i(i=1,2,\cdots,s)$之间的关系,再代回$A_{m\times n}x=0$的通解中就得到公共解。 \item 给出$A_{m\times n}x=0$的通解与$B_{m\times n}x=0$的通解联立:$k_1\xi_1+k_2\xi_2+\cdots+k_s\xi_s=l_1\eta_1+l_2\eta_2+\cdots+l_s\eta_s=0$,能解出$k_i$和$l_i$。 \end{enumerate} +这种方法可以求出公共解,不过比较麻烦。 + +如果已经给出原方程的基础解系而没有给出矩阵,则这个方法解出公共解较好。 + \textbf{例题:}已知线性方程组$A=\left\{\begin{array}{l} x_1+x_2=0 \\ x_2-x_4=0 @@ -539,12 +545,29 @@ $l_1\eta_1+l_2\eta_2=l_1(0,1,1,0)^T+l_2(-1,-1,0,1)^T=(-l_2,l_1-l_2,l_1,l_2)^T$ 公共解为$(-k_2,k_2,2k_2,k_2)^T=k_2(-1,1,2,1)^T$。 +\subsection{矩阵法} + +要求$A$和$B$的非零公共解,即求联立矩阵$\left(\begin{array}{c} + A \\ + B +\end{array}\right)x=0$的非零解。对这个矩阵求出基础解系。 + +如果直接给出矩阵,则这种方法可以不用求出基础解系就能得到公共解。 + \section{同解方程组} +\subsection{性质} + 若$A_{m\times n}x=0$和$B_{s\times n}x=0$有完全相同的解,就是同解方程组。 $\therefore r(A)=r(B)=r([A,B]^T)$。 +$A$与$A^TA$同解。 + +\subsection{代入法} + +先求一个方程组的通解,然后把这个通解代入到第二个方程组中,不用管$k$的取值(因为$k$为任意数,所以直接令其为0)直接求出对应参数。 + \textbf{例题:}线性方程组$A=\left\{\begin{array}{l} x_1+3x_3+5x_4=0 \\ x_1-x_2-2x_3+2x_4=0 \\