Two Subspaces Being Equal I know that having the same dimension is one of the conditions for 2 subspaces being equal. What other conditions do I need to check to see if 2 subspaces (null space and column space) are equal?
 A: Testing if two arbitrary subspaces are equal is one thing, but testing if the null space and the column space of a matrix $A$ are equal is a much more specialized question. I'll answer that question.
The column space of a matrix $A$ is the range of the matrix: the set of all places it could send a vector, i.e. $\{y \mid \mathbf{A}x = y\}$.
The null space of a matrix is the set of all vectors it kills: everything it sends to $0$, i.e. $\{x \mid \mathbf{A}x = 0\}$.
If the null space equals the column space, that means a few things:


*

*the matrix must be $n \times n$ (because you're applying it to its own outputs)

*$n$ must be an even number. By of the rank-nullity theorem: a matrix acting on $\mathbb{R}^n$ must have (number of dimensions it kills) + (number of dimensions it preserves) equal to $n$. $\mathbf{\dim}{(\text{column space})}$ is the number of dimensions it preserves (its rank) and $\mathbf{\dim}{(\text{null space})}$ is the number of dimensions it kills.

*the matrix must be nilpotent (since $\mathbf{A}y = 0 \implies \mathbf{A}(\mathbf{A}x) = 0 \implies \mathbf{A}^2x = 0$)


and additional conditions are given here.
