I have been working practice problems that give conditions if a map is bijective, surjective, etc. and this could be considered a follow up to this post Condition on minors of a matrix to check when linear transformation is injective but I do not think they are related other than they are from the same family of similar facts I am trying to uncatalogued for a test. I
Let $V,W$ be two finite dimensional vectors spaces. Suppose that $f: V \rightarrow W$ is a linear mapping and let $A$ be the matrix corresponding to the linear map $f$.
How do you show if $\dim V \geq \dim W$ and there exists a minor of $A$ with the same of order as $\dim W$, then the map $f$ is surjective.