Question

在p^4中,求由基ξ1,ξ2,ξ3,ξ4到基η1,η2,η3,η4的过渡矩阵,并求向量α在所指基底下的坐标.

Answer 1

(ξ1,ξ2,ξ3,ξ4,η1,η2,η3,η4)=

1 1 1 1 1 2 1 0

1 1 -1 -1 1 1 1 1

1 -1 1 -1 0 3 0 -1

1 -1 -1 1 1 1 0 -1

ri-r1,i=2,3,4

1 1 1 1 1 2 1 0

0 0 -2 -2 0 -1 0 1

0 -2 0 -2 -1 1 -1 -1

0 -2 -2 0 0 -1 -1 -1

r4-r2-r3

1 1 1 1 1 2 1 0

0 0 -2 -2 0 -1 0 1

0 -2 0 -2 -1 1 -1 -1

0 0 0 4 1 -1 0 -1

r4*(-1/4), r1-r4,r2+2r4,r3+2r4

1 1 1 0 3/4 9/4 1 1/4

0 0 -2 0 2/4 -6/4 0 2/4

0 -2 0 0 -2/4 2/4 -1 -6/4

0 0 0 1 1/4 -1/4 0 -1/4

r2*(-1/2),r3*(-1/2)

1 1 1 0 3/4 9/4 1 1/4

0 0 1 0 -1/4 3/4 0 -1/4

0 1 0 0 1/4 -1/4 1/2 3/4

0 0 0 1 1/4 -1/4 0 -1/4

r1-r2-r3, r2r3

1 0 0 0 3/4 7/4 1/2 -1/4

0 1 0 0 1/4 -1/4 1/2 3/4

0 0 1 0 -1/4 3/4 0 -1/4

0 0 0 1 1/4 -1/4 0 -1/4

过渡矩阵为

3/4 7/4 1/2 -1/4

1/4 -1/4 1/2 3/4

-1/4 3/4 0 -1/4

1/4 -1/4 0 -1/4