Onto transformation properties

Question

If $A \in \mathbb{R}^{\ n m}$ is a matrix, and $f(x) = Ax$ the corresponding map.

Can we say that if $f$ is onto, then the system $Ax = b$ always has at least one solution regardless of b?

Answer 1

Yes. This is the definition of onto: for all $b\in \mathbb{R}^m$ there is $x \in \mathbb{R}^n$ such that $f(x) = b$ - that is, $Ax=b$