高等代数

高等代数理论基础40：基变换与坐标变换

2019-02-06  本文已影响11人  溺于恐

基变换与坐标变换

设\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n​与\varepsilon'_1,\varepsilon'_2,\cdots,\varepsilon'_n​是n维线性空间V中两组基,它们有如下关系

\begin{cases}\varepsilon'_1=a_{11}\varepsilon_1+a_{21}\varepsilon_2+\cdots+a_{n1}\varepsilon_n\\ \varepsilon'_2=a_{12}\varepsilon_1+a_{22}\varepsilon_2+\cdots+a_{n2}\varepsilon_n\\ \cdots\\ \varepsilon'_n=a_{1n}\varepsilon_1+a_{2n}\varepsilon_2+\cdots+a_{nn}\varepsilon_n\end{cases}

设向量\xi在这两组基下的坐标分别为(x_1,x_2,\cdots,x_n)与(x'_1,x'_2,\cdots,x'_n),即\xi=x_1\varepsilon_1+x_2\varepsilon_2+\cdots+x_n\varepsilon_n,\xi=x'_1\varepsilon'_1+x'_2\varepsilon'_2+\cdots+x'_n\varepsilon'_n

(a_{1j},a_{2j},\cdots,a_{nj}),j=1,2,\cdots,n为基向量\varepsilon'_j(j=1,2,\cdots,n)在第一组基下的坐标,向量\varepsilon'_1,\varepsilon'_2,\cdots,\varepsilon'_n线性无关保证方程组系数矩阵的行列式不为零,即系数矩阵可逆

\xi=x_1\varepsilon_1+x_2\varepsilon_2+\cdots+x_n\varepsilon_n​

=(\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n)\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}​

(将基写成1\times n矩阵,将坐标写成n\times 1矩阵)

(\varepsilon'_1,\varepsilon'_2,\cdots,\varepsilon'_n)=(\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n)\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots& &\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}​

设A=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots& &\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}

A称为由基\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n到\varepsilon'_1,\varepsilon'_2,\cdots,\varepsilon'_n的过渡矩阵,A可逆

运算规律

设\alpha_1,\alpha_2,\cdots,\alpha_n和\beta_1,\beta_2,\cdots,\beta_n是V中两个向量组,A=(a_{ij})_{n\times n},B=(b_{ij})_{n\times n},则

1.((\alpha_1,\alpha_2,\cdots,\alpha_n)A)B=(\alpha_1,\alpha_2,\cdots,\alpha_n)(AB)

2.(\alpha_1,\alpha_2,\cdots,\alpha_n)A+(\alpha_1,\alpha_2,\cdots,\alpha_n)B

=(\alpha_1,\alpha_2,\cdots,\alpha_n)(A+B)

3.(\alpha_1,\alpha_2,\cdots,\alpha_n)A+(\beta_1,\beta_2,\cdots,\beta_n)A

=(\alpha_1+\beta_1,\alpha_2+\beta_2,\cdots,\alpha_n+\beta_n)A

\xi=(\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n)\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}​

\xi=(\varepsilon'_1,\varepsilon'_2,\cdots,\varepsilon'_n)\begin{pmatrix}x'_1\\x'_2\\\vdots\\x'_n\end{pmatrix}​

故\xi=(\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n)\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots& &\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}\begin{pmatrix}x'_1\\x'_2\\\vdots\\x'_n\end{pmatrix}​

由基向量的线性无关性

\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots& &\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}\begin{pmatrix}x'_1\\x'_2\\\vdots\\x'_n\end{pmatrix}​

或\begin{pmatrix}x'_1\\x'_2\\\vdots\\x'_n\end{pmatrix}=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots& &\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}^{-1}\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}

上式即为基变换下向量坐标变换公式

例：在n维空间P^n中,\begin{cases}\varepsilon_1=(1,0,\cdots,0)\\ \varepsilon_2=(0,1,\cdots,0)\\ \cdots\\ \varepsilon_n=(0,0,\cdots,1)\end{cases}为一组基,

\begin{cases}\varepsilon'_1=(1,1,\cdots,1)\\ \varepsilon'_2=(0,1,\cdots,1)\\ \cdots\\ \varepsilon'_n=(0,0,\cdots,1)\end{cases}​为另一组基

(\varepsilon'_1,\varepsilon'_2,\cdots,\varepsilon'_n)=(\varepsilon_1,\varepsilon_2,\cdots,\varepsilon_n)\begin{pmatrix}1&0&\cdots&0\\ 1&1&\cdots&0\\ \vdots&\vdots& &\vdots\\ 1&1&\cdots&1\end{pmatrix}

A=\begin{pmatrix}1&0&\cdots&0\\ 1&1&\cdots&0\\ \vdots&\vdots& &\vdots\\ 1&1&\cdots&1\end{pmatrix}

A^{-1}=\begin{pmatrix}1&0&0&\cdots&0\\ -1&1&0&\cdots&0\\ 0&-1&1&\cdots&0\\ \vdots&\vdots&\vdots& &\vdots\\ 0&0&0&\cdots&1\end{pmatrix}​

故\begin{pmatrix}x'_1\\x'_2\\\vdots\\x'_n\end{pmatrix}=\begin{pmatrix}1&0&0&\cdots&0\\ -1&1&0&\cdots&0\\ 0&-1&1&\cdots&0\\ \vdots&\vdots&\vdots& &\vdots\\ 0&0&0&\cdots&1\end{pmatrix}\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}

即x'_1=x_1,x'_i=x_i-x_{i-1}(i=2,\cdots,n)

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