# Checking that a matrix space is invariant with respect to a linear map

I am trying to solve Checkpoint 1.4.4 from "A Second Course in Linear Algebra" from Robert A. Beezer. I need to prove that V is invariant with respect to B.

But I think it is not actually true. In order to prove it, I am comparing the rank(V) = 3 and the rank of (V | BV) which is 4. So there must be a vector in BV that is linearly independent with the vectors of V. Do I miss something?

Here is my code:

```
B = matrix (
[[4 , 47 , 3 , -46 , 20] ,
[10 , 61 , 8 , -56 , 10] ,
[ -10 , -69 , -7 , 67 , -20] ,
[11 , 70 , 9 , -64 , 12] ,
[3 , 19 , 3 , -16 , 1]])
V = matrix([[1, 3, -2],
[1, 3, 3],
[-1, -2, 2],
[1, 4, 3],
[0, 2, 1]])
V.augment(B*V).rank() == V.rank()
```

Your reasoning sounds fine to me. It could be a typo somewhere.