高等代数

高等代数理论基础46：线性变换的运算

2019-03-26  本文已影响3人  溺于恐

线性变换的运算

乘法

设\mathscr{A},\mathscr{B}是线性空间V的两个线性变换,定义它们的乘积\mathscr{A}\mathscr{B}为(\mathscr{A}\mathscr{B})(\alpha)=\mathscr{A}(\mathscr{B}(\alpha))(\alpha\in V)

性质：

1.线性变换的乘积也是线性变换

(\mathscr{A}\mathscr{B})(\alpha+\beta)=\mathscr{A}(\mathscr{B}(\alpha+\beta))

=\mathscr{A}(\mathscr{B}(\alpha)+\mathscr{B}(\beta))​

=\mathscr{A}(\mathscr{B}(\alpha))+\mathscr{A}(\mathscr{B}(\beta))​

=(\mathscr{AB})(\alpha)+(\mathscr{AB})(\beta)

(\mathscr{AB})(k\alpha)=\mathscr{A}(\mathscr{B}(k\alpha))

=\mathscr{A}(k\mathscr{B}(\alpha))=k\mathscr{A}(\mathscr{B}(\alpha))=k(\mathscr{AB})(\alpha)

即\mathscr{AB}是线性的

2.线性变换的乘法适合结合律

(\mathscr{A}\mathscr{B})\mathscr{C}=\mathscr{A}(\mathscr{B}\mathscr{C})​

3.线性变换的乘法一般不可交换

例：\R上的线性空间\R[x]中,线性变换

\mathscr{D}(f(x))=f’(x),\mathscr{G}(f(x))=\int_0^xf(t)dt

\mathscr{DG}=\mathscr{E}​,但一般\mathscr{GD}\neq\mathscr{E}​

4.对任意线性变换\mathscr{A},\mathscr{EA=AE=A}

逆变换

对V的变换\mathscr{A},若有V的变换\mathscr{B},使\mathscr{AB=BA=E}

则称变换\mathscr{A}可逆,变换\mathscr{B}称为\mathscr{A}的逆变换,记作\mathscr{A^{-1}}

若线性变换\mathscr{A}是可逆的,则它的逆变换\mathscr{A^{-1}}也是线性变换

\mathscr{A^{-1}(\alpha+\beta)=A^{-1}[(AA^{-1})(\alpha)+(AA^{-1})(\beta)]}​

=\mathscr{A^{-1}[A(A^{-1}(\alpha))+A(A^{-1}(\beta))]}

=\mathscr{A^{-1}[A(A^{-1}(\alpha)+A^{-1}(\beta))]}

=\mathscr{(A^{-1}A)(A^{-1}(\alpha)+A^{-1}(\beta))}​

=\mathscr{A^{-1}(\alpha)+A^{-1}(\beta)}

\mathscr{A^{-1}(k\alpha)=A^{-1}(k(AA^{-1})(\alpha))}

=\mathscr{A^{-1}(k(A(A^{-1}(\alpha))))}

=\mathscr{A^{-1}(A(kA^{-1}(\alpha)))}

=\mathscr{(A^{-1}A)(kA^{-1}(\alpha))}

=\mathscr{kA^{-1}(\alpha)}

即\mathscr{A^{-1}}是线性变换

加法

设\mathscr{A,B}​是线性空间V的两个线性变换,定义它们的和\mathscr{A+B}​为

\mathscr{(A+B)(\alpha)=A(\alpha)+B(\alpha)})(\alpha \in V)​

性质：

1.线性变换的和还是线性变换

\mathscr{(A+B)}(\alpha+\beta)=\mathscr{A}(\alpha+\beta)+\mathscr{B}(\alpha+\beta)

=(\mathscr{A}(\alpha)+\mathscr{A}(\beta))+(\mathscr{B}(\alpha)+\mathscr{B}(\beta))

=\mathscr{(A(\alpha)+B(\alpha))+(A(\beta)+B(\beta))}

=\mathscr{(A+B)(\alpha)+(A+B)(\beta)}

\mathscr{(A+B)(k\alpha)=A(k\alpha)+B(k\alpha)}

=\mathscr{kA(\alpha)+kB(\alpha)}

=\mathscr{k(A(\alpha)+B(\alpha))}​

=k\mathscr{(A+B)(\alpha)}​

即\mathscr{A+B}是线性变换

2.线性变换的加法适合结合律与交换律

\mathscr{A+(B+C)}=\mathscr{(A+B)+C}

\mathscr{A+B=B+A}

3.对任意线性变换\mathscr{A},\mathscr{A+O=A}

4.分配律

左分配律\mathscr{A(B+C)=AB+AC}

右分配律\mathscr{(B+C)A=BA+CA}

\mathscr{A(B+C)(\alpha)=A((B+C)(\alpha))}​

=\mathscr{A(B(\alpha)+C(\alpha))}​

=\mathscr{A(B(\alpha))+A(C(\alpha))}​

=\mathscr{(AB)(\alpha)+(AC)(\alpha)}​

=\mathscr{(AB+AC)(\alpha)}

负变换

对任意线性变换\mathscr{A}​,定义负变换-\mathscr{A}​

\mathscr{(-A)(\alpha)}=-\mathscr{A}(\alpha)(\alpha\in V)

负变换也是线性的,且\mathscr{A+(-A)}=O

数量乘法

定义数域P中的数与线性变换的数量乘法

k\mathscr{A=KA}

即\mathscr{(kA)(\alpha)=K(A(\alpha))=KA(\alpha)}

注：k\mathscr{A}还是线性变换

规律：

(kl)\mathscr{A}=k(l\mathscr{A})

(k+l)\mathscr{A}=k\mathscr{A}+l\mathscr{A}

k(\mathscr{A+B})=k\mathscr{A}+k\mathscr{B}

1\mathscr{A}=\mathscr{A}

注：线性空间V上全体线性变换,对如上定义的加法与数量乘法,构成数域P上一个线性空间

线性变换的乘幂

n个线性变换\mathscr{A}相乘,称为\mathscr{A}的n次幂,简记作\mathscr{A^n}

定义\mathscr{A^0=E}

指数法则：

\mathscr{A^{m+n}}=\mathscr{A^mA^n}

\mathscr{(A^m)^n=A^{mn}}(m,n\ge 0)

\mathscr{A}可逆时,\mathscr{A^{-n}=(A^{-1})^n},n\in Z_+

注：线性变换乘积的指数法则不成立

一般,\mathscr{(AB)^n\neq A^nB^b}

线性变换的多项式

设f(x)=a_mx^m+a_{m-1}x^{m-1}+\cdots+a_0\in P[x]

\mathscr{A}是V的一线性变换

定义f(\mathscr{A})=a_m\mathscr{A}^m+a_{m-1}\mathscr{A}^{m-1}+\cdots+a_0\mathscr{E}

显然f(\mathscr{A})是一线性变换,称为线性变换\mathscr{A}的多项式

易证,若P[x]中h(x)=f(x)+g(x),p(x)=f(x)g(x)

则h(\mathscr{A})=f(\mathscr{A})+g(\mathscr{A}),p(\mathscr{A})=f(\mathscr{A})g(\mathscr{A})​

同一个线性变换的多项式的乘法可交换

f(\mathscr{A})g(\mathscr{A})=g(\mathscr{A})f(\mathscr{A})​

例：

1.线性空间P[\lambda]_n中,求微商是一个线性变换\mathscr{D^n=O}

平移f(\lambda)\to f(\lambda+a)(a\in P)是一个线性变换\mathscr{G}_a

由泰勒展开式f(\lambda+a)=f(\lambda)+af’(\lambda)+{a^2\over 2!}f’‘(\lambda)+\cdots+{a^{n-1}\over (n-1)!}f^{(n-1)}(\lambda)

故\mathscr{G}_a是\mathscr{D}的多项式

\mathscr{G}_a=\mathscr{E}+a\mathscr{D}+{a^2\over 2!}\mathscr{D}^2+\cdots+{a^{n-1}\over (n-1)!}\mathscr{D}^{n-1}

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