If two matrices have the same column space and null space, are they the same matrix? If two matrices have the same column space and null space, are they the same matrix? I am thinking no because if A=[1 2;2 1] and B=[2 1;1 2] then they have the same column space (I think) but they are not identical

This fails even in one dimension: $1$ and $2$ have the same column and null spaces. You can easily find other examples in higher dimensions. For example $I$ and $2I$.
In fact, all invertible matrices have the same column and null spaces, yet there are many different invertible matrices.
