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?
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I guess "equal" really means "isomorphic" here? – MPW Jan 12 '16 at 18:14
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2If there is a common basis they are equal. – Emilio Novati Jan 12 '16 at 18:15
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Also see this: http://math.stackexchange.com/questions/1532389/how-to-prove-that-two-subspaces-are-equal-if-and-only-if-their-annihilators-are – David K Jan 12 '16 at 18:17
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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.
