# Column space and null space

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?

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 $$\mathrm{nullity}=2$$.