### 15.1.7 Equivalent matrix

The changebase command changes a matrix to represent the same linear function in a different basis.

- changebase takes two arguments:
  - A, a matrix.
  - P, a change-of-basis matrix.
- changebase(A,P) returns the matrix P−1AP.

#### Examples

changebase([[1,2],[3,4]],[[1,0],[0,1]])

⎡
⎢
⎣

1 | 2
3 | 4

⎤
⎥
⎦

changebase([[1,1],[0,1]],[[1,2],[3,4]])

⎡
⎢
⎢
⎢
⎢
⎣

−5 | −8

9
2

7

⎤
⎥
⎥
⎥
⎥
⎦

Indeed:

⎡
⎢
⎣

1 | 2
3 | 4

⎤
⎥
⎦

−1

·

⎡
⎢
⎣

1 | 1
0 | 1

⎤
⎥
⎦

·

⎡
⎢
⎣

1 | 2
3 | 4

⎤
⎥
⎦

=

⎡
⎢
⎢
⎢
⎢
⎣

−5 | −8

9
2

7

⎤
⎥
⎥
⎥
⎥
⎦

.