子节 3.3.1 主要知识点
定义 3.3.1.
设\(\xi_1,\xi_2,\cdots,\xi_n\)是线性空间\(V\)的一个基，对于任意的向量\(\alpha\in V\)，有
\begin{equation*}
\alpha=a_1\xi_1+a_2\xi_2+\cdots +a_n\xi_n,
\end{equation*}
这里的\(a_1,a_2,\cdots,a_n\in \mathbb{F}\)是唯一确定的， 将\((a_1,a_2,\cdots,a_n)^T\)称为\(\alpha\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下的坐标，形式上记作
\begin{equation*}
\alpha=(\xi_1,\xi_2,\cdots,\xi_n)\left(\begin{array}{c}
a_1\\a_2\\\vdots\\a_n
\end{array}\right).
\end{equation*}
命题 3.3.2.
对\(V\)中任意向量\(\alpha,\beta\)， 若它们在基\(\xi_1,\xi_2,\cdots,\xi_n\)下坐标为\(X,Y\)， 则对任意\(c\in \mathbb{F}\)，
\(\alpha+\beta\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下坐标为\(X+Y\)；
\(c\alpha\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下坐标为\(cX\)。
例 3.3.3.
设\(\xi_1,\xi_2,\cdots,\xi_n\)是线性空间\(V\)的一个基，求向量\(\xi_i\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下的坐标，\(i=1,2,\cdots,n\)。
例 3.3.4.
已知\(A=\left(\begin{array}{cc}
2&1\\-1&0
\end{array}\right)\)
求\(A\)在基\(E_{11},E_{12},E_{21},E_{22}\)下的坐标；
求\(A\)在基\(E_{12},E_{11},E_{21},E_{22}\)下的坐标；
求\(A\)在基\(E_{11},E_{12},E_{11}+E_{21},E_{22}\)下的坐标。
向量的形式书写法
设线性空间\(V\)中向量组 \(\beta_1,\beta_2,\cdots,\beta_t\)可由向量组 \(\alpha_1,\alpha_2,\cdots,\alpha_s\) 线性表出，即存在\(a_{ij}\in \mathbb{F}, \ i=1,2,\cdots s,\ j=1,2,\cdots,t\)，使得
\begin{equation*}
\left\{\begin{array}{c}
\beta_1=a_{11}\alpha_1+a_{21}\alpha_2+\cdots+a_{s1}\alpha_s\\
\beta_2=a_{12}\alpha_1+a_{22}\alpha_2+\cdots+a_{s2}\alpha_s\\
\cdots\\
\beta_t=a_{1t}\alpha_1+a_{2t}\alpha_2+\cdots+a_{st}\alpha_s\\
\end{array}\right. ,
\end{equation*}
形式上记作
\begin{equation*}
(\beta_1,\beta_2,\cdots,\beta_t)=(\alpha_1,\alpha_2,\cdots,\alpha_s)
\left(\begin{array}{cccc}
a_{11}&a_{12}&\cdots&a_{1t}\\
a_{21}&a_{22}&\cdots&a_{2t}\\
\cdots&\cdots&\cdots&\cdots\\
a_{s1}&a_{s2}&\cdots&a_{st}
\end{array}\right),
\end{equation*}
即存在\(A\in \mathbb{F}^{s\times t}\)，使得
\begin{equation*}
(\beta_1,\beta_2,\cdots,\beta_t)=(\alpha_1,\alpha_2,\cdots,\alpha_s)A.
\end{equation*}
设\(\alpha_1,\alpha_2,\cdots,\alpha_s\in V,\ A,\ B\in \mathbb{F}^{s\times t},\ C\in \mathbb{F}^{t\times p}\)，则
\begin{equation*}
(\alpha_1,\alpha_2,\cdots,\alpha_s)A+(\alpha_1,\alpha_2,\cdots,\alpha_s)B=(\alpha_1,\alpha_2,\cdots,\alpha_s)(A+B);
\end{equation*}
\begin{equation*}
c[(\alpha_1,\alpha_2,\cdots,\alpha_s)A]=(\alpha_1,\alpha_2,\cdots,\alpha_s)(cA);
\end{equation*}
\begin{equation*}
\left((\alpha_1,\alpha_2,\cdots,\alpha_s)B\right)C=(\alpha_1,\alpha_2,\cdots,\alpha_s)(BC).
\end{equation*}
设\(\alpha_1,\alpha_2,\cdots,\alpha_s\)线性无关，则
\begin{equation*}
(\alpha_1,\alpha_2,\cdots,\alpha_s)A=(\alpha_1,\alpha_2,\cdots,\alpha_s)B\Leftrightarrow A=B.
\end{equation*}
命题 3.3.6.
设\(V\)是\(n\)维线性空间，\(\xi_1,\xi_2,\cdots,\xi_n\)是\(V\)的一个基，
对\(V\)中任意向量\(\alpha,\beta\)， 若它们在基\(\xi_1,\xi_2,\cdots,\xi_n\)下坐标为\(X,Y\)， 则对任意\(c\in \mathbb{F}\)， \(\alpha+\beta,\ c\alpha\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下坐标为\(X+Y,\ cX\)；
若\(V\)中向量\(\alpha_1,\alpha_2,\cdots,\alpha_s\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下坐标分别为\(X_1,X_2,\cdots,X_s\)，则\(\alpha_1,\alpha_2,\cdots,\alpha_s\)线性相关的充分必要条件是\(X_1,X_2,\cdots,X_s\)线性相关。
定义 3.3.7.
设\(\xi_1,\xi_2,\cdots,\xi_n\)和\(\eta_1,\eta_2,\cdots,\eta_n\)是线性空间\(V\)的两个基。若
\begin{equation*}
\left\{\begin{array}{c}
\eta_1=a_{11}\xi_1+a_{21}\xi_2+\cdots+a_{n1}\xi_n\\
\eta_2=a_{12}\xi_1+a_{22}\xi_2+\cdots+a_{n2}\xi_n\\
\cdots\\
\eta_n=a_{1n}\xi_1+a_{2n}\xi_2+\cdots+a_{nn}\xi_n\\
\end{array}\right. ,
\end{equation*}
即
\begin{equation*}
(\eta_1,\eta_2,\cdots,\eta_n)=(\xi_1,\xi_2,\cdots,\xi_n)A,
\end{equation*}
则称
\begin{equation*}
A=\left(\begin{array}{cccc}
a_{11}&a_{12}&\cdots&a_{1n}\\
a_{21}&a_{22}&\cdots&a_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
a_{n1}&a_{n2}&\cdots&a_{nn}
\end{array}\right)
\end{equation*}
是基\(\xi_1,\xi_2,\cdots,\xi_n\)到基\(\eta_1,\eta_2,\cdots,\eta_n\)的过渡矩阵。
命题 3.3.8.
设\(\xi_1,\xi_2,\cdots,\xi_n\)是\(n\)维线性空间\(V\)的一个基，\(\eta_1,\eta_2,\cdots,\eta_n;\)\ \(\zeta_1,\zeta_2,\cdots,\zeta_n\)是\(V\)的另两个基， 且
\begin{equation*}
\begin{array}{c}
(\eta_1,\eta_2,\cdots,\eta_n)=(\xi_1,\xi_2,\cdots,\xi_n)A,\\
(\zeta_1,\zeta_2,\cdots,\zeta_n)=(\eta_1,\eta_2,\cdots,\eta_n)B,
\end{array}
\end{equation*}
则
\begin{equation*}
(\zeta_1,\zeta_2,\cdots,\zeta_n)=(\xi_1,\xi_2,\cdots,\xi_n)AB.
\end{equation*}
即若\(\xi_1,\xi_2,\cdots,\xi_n\)到\(\eta_1,\eta_2,\cdots,\eta_n\)的过渡矩阵为\(A\)，\(\eta_1,\eta_2,\cdots,\eta_n\)到\(\zeta_1,\zeta_2,\cdots,\zeta_n\)的过渡矩阵为\(B\)，则\(\xi_1,\xi_2,\cdots,\xi_n\)到\(\zeta_1,\zeta_2,\cdots,\zeta_n\)的过渡矩阵为\(AB\)。
命题 3.3.9.
设\(\xi_1,\xi_2,\cdots,\xi_n\)和\(\eta_1,\eta_2,\cdots,\eta_n\)是\(n\)维线性空间\(V\)的两个基，且
\begin{equation*}
(\eta_1,\eta_2,\cdots,\eta_n)=(\xi_1,\xi_2,\cdots,\xi_n)A,\quad
(\xi_1,\xi_2,\cdots,\xi_n)=(\eta_1,\eta_2,\cdots,\eta_n)B,
\end{equation*}
则\(A\)可逆且\(A^{-1}=B\)。
命题 3.3.10.
设\(\xi_1,\xi_2,\cdots,\xi_n\)和\(\eta_1,\eta_2,\cdots,\eta_n\)是\(n\)维线性空间\(V\)的两个向量组，且
\begin{equation*}
(\eta_1,\eta_2,\cdots,\eta_n)=(\xi_1,\xi_2,\cdots,\xi_n)A,
\end{equation*}
若\(\xi_1,\xi_2,\cdots,\xi_n\)和\(\eta_1,\eta_2,\cdots,\eta_n\)是\(V\)的两个基，则\(A\)可逆。
若\(\xi_1,\xi_2,\cdots,\xi_n\)是\(V\)的一个基且\(A\)可逆，则\(\eta_1,\eta_2,\cdots,\eta_n\)是\(V\)的一个基。
若\(\eta_1,\eta_2,\cdots,\eta_n\)是\(V\)的一个基且\(A\)可逆，则\(\xi_1,\xi_2,\cdots,\xi_n\)是\(V\)的一个基。
例 3.3.11.
在\(\mathbb{F}_3\)中， 已知
\begin{equation*}
\begin{array}{l}
{\xi_1} = (1,0, - 1),{\xi_2} = (2,1,1),{\xi_3} = (1,1,1);\\
{\eta_1} = (1,2,1),{\eta_2} = (1,2,0),{\eta_3} = (1,0,0).
\end{array}
\end{equation*}
求从基\(\xi_1,\xi_2,\xi_3\)到\(\eta_1,\eta_2,\eta_3\)的过渡矩阵。
例 3.3.12.
设
\begin{equation*}
\xi_1=\left(\begin{array}{cc}
1&1\\0&1
\end{array}\right),\xi_2=\left(\begin{array}{cc}
2&1\\3&1
\end{array}\right),\xi_3=\left(\begin{array}{cc}
1&1\\0&0
\end{array}\right),\xi_4=\left(\begin{array}{cc}
0&1\\-1&-1
\end{array}\right),
\end{equation*}
证明：\(\xi_1,\xi_2,\xi_3,\xi_4\)是\(\mathbb{R}^{2\times 2}\)的一个基。
设\(\xi_1,\xi_2,\cdots,\xi_n\)和\(\eta_1,\eta_2,\cdots,\eta_n\)是线性空间\(V\)的两个基，且
\begin{equation*}
(\eta_1,\eta_2,\cdots,\eta_n)=(\xi_1,\xi_2,\cdots,\xi_n)A,
\end{equation*}
若\(\alpha\in V\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)与\(\eta_1,\eta_2,\cdots,\eta_n\)下的坐标分别为\(X,Y\)，即
\begin{equation*}
\alpha=(\xi_1,\xi_2,\cdots,\xi_n)X,\ \ \alpha=(\eta_1,\eta_2,\cdots,\eta_n)Y,
\end{equation*}
则
\begin{equation*}
\alpha=[(\xi_1,\xi_2,\cdots,\xi_n)A]Y=(\xi_1,\xi_2,\cdots,\xi_n)(AY).
\end{equation*}
因此，
\begin{equation*}
X=AY.
\end{equation*}
命题 3.3.13.
设线性空间\(V\)的基\(\xi_1,\xi_2,\cdots,\xi_n\)到\(\eta_1,\eta_2,\cdots,\eta_n\)的过渡矩阵为\(A\)，若向量\(\alpha\)在基\(\eta_1,\eta_2,\cdots,\eta_n\)下的坐标为\(X\)，则\(\alpha\)在基\(\xi_1,\xi_2,\cdots,\xi_n\)下的坐标为\(AX\)。
例 3.3.14.
设
\begin{equation*}
\xi_1=\left(\begin{array}{cc}
1&1\\0&1
\end{array}\right),\xi_2=\left(\begin{array}{cc}
2&1\\3&1
\end{array}\right),\xi_3=\left(\begin{array}{cc}
1&1\\0&0
\end{array}\right),\xi_4=\left(\begin{array}{cc}
0&1\\-1&-1
\end{array}\right),
\end{equation*}
求\(\alpha=\left(\begin{array}{cc}
1&0\\0&0
\end{array}\right)\)在基\(\xi_1,\xi_2,\xi_3,\xi_4\)下的坐标。
