Question

矩阵习题。一定要过程，谢谢了

线性空间V为由基函数x1=(e^at)cos(bt), x2=(e^at)sin(bt), x3=t(e^at)cos(bt),
x4=t(e^at)sin(bt), x5={t^2[(e^at)cos(bt)]}/2 , x6={t^2[(e^at)sin(bt)]}/2
生成的实数域上的线性空间，求微分算子D在这组基下的矩阵

Answer 1

微分算子D在这组基下的变换为：
Dx1= ax1 - bx2
Dx2= bx1 + ax2
Dx3= x1 + ax3 - bx4
Dx4= x2 + bx3 + ax4
Dx5= x3 + ax5 - bx6
Dx6= x4 + bx5 + ax6

设微分算子D在这组基下的矩阵为A：
D(x1,x2,x3,x4,x5,x6) = (x1,x2,x3,x4,x5,x6)A

A =
a b 1 0 0 0
-b a 0 1 0 0
0 0 a b 1 0
0 0 -b a 0 1
0 0 0 0 a b
0 0 0 0 -b a