3.5 基变换与过渡矩阵
42. 设$\{u_{1},u_{2},\cdots,u_{n}\}$, $\{e_{1},e_{2},\cdots,e_{n}\}$, $\{f_{1},f_{2},\cdots,f_{n}\}$是向量空间$V$的三组基，若从$\{u_{1},u_{2},\cdots,u_{n}\}$到$\{e_{1},e_{2},\cdots,e_{n}\}$的过渡矩阵是A，以及从$\{u_{1},u_{2},\cdots,u_{n}\}$到$\{f_{1},f_{2},\cdots,f_{n}\}$的过渡矩阵是B，那么问题来啦，求从$\{e_{1},e_{2},\cdots,e_{n}\}$到$\{f_{1},f_{2},\cdots,f_{n}\}$的过渡矩阵
43. 求四维行向量空间中$e_{1},e_{2},\cdots,e_{n}$到$f_{1},f_{2},\cdots,f_{n}$的过渡矩阵，其中
$$ e_{1}=(1101),e_{2}=(2120),e_{3}=(1100),e_{4}=(0,1,-1,-1) $$ $$ f_{1}=(1001),f_{2}=(0,0,1,-1),f_{3}=(2103),f_{4}=(-1,0,1,2) $$
44. 设a为常数，求向量$\alpha=(a_{1},a_{2},\cdots,a_{n})$在基
$$\{f_{1}=(a^{n-1}),a^{n-2},\cdots,a,1\} \{f_{2}=(a^{n-2}),a^{n-3},\cdots,1,0\} \{f_{n}=(1,0,\cdots,0,0)\} $$ 下的坐标
45. 设$V$是次数不超过n的实系数多项式全体组成的线性空间，求从基$\{1,x,x^{2},\cdots,x^{n}\}$到基$\{1,(x-a),(x-a)^{2},\cdots,(x-a)^{n}\}$的过渡矩阵，并以此证明多项式的Taylor公式
$$ f(x)=f(a)+\frac{f’(a)}{1!}(x-a)+\frac{f^{2}(a)}{2!}(x-a)^(2)+\cdots+\frac{f^{n}(a)}{n!}(x-a)^{n} $$