TITLE: Column space and null space
QUESTION [5 upvotes]: Let $A\in M_{5,7}(\mathbb{R})$ be a matrix such that $Ax=b$ has solution for every $b$.

I have to say what this information tells me about column- and null-space and rows of a matrix. The only thing I can think of is that column-space can have dimension $1$ to $5$ and null-space dimension can be deduced using rank-nullity theorem.
Is there something more to see here?

REPLY [5 votes]: Column space is all of $\mathbb{R}^5$ since any $b\in \mathbb{R}^5$ appears in the range of $A$.   Thus rank is 5, and so the rank-nullity theorem says $5+\mathrm{nullity}=7$, thus $\mathrm{nullity}=2$.