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
2026-05-15 16:01:20.1778860880
If two matrices have the same column space and null space, are they the same matrix?
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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.