5
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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 $\mathrm{nullity}=2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.