TITLE: Matrix characterization of surjective and injective linear functions
QUESTION [6 upvotes]: I don't remember well of my Linear Algebra classes, looking the rank of a matrix $A\in M(n\times m)$ how can we say the application associated to this matrix is surjective or injective?
For a matrix $A \in M(n\times n)$ is easy, $A$ is surjective (injective) iff $\text{rank} A=n$ .
Thanks

REPLY [11 votes]: The rank-nullity theorem is the key here. If your matrix $A$ is $n\times m$, with rank $k$, the theorem says that
$$
\dim \ker f=m-k
$$
where $f$ is the linear map with associated matrix $A$.
Injectivity is equivalent to $\dim\ker f=0$ or, by the theorem, to $k=m$, while surjectivity is equivalent to $k=n$.
Since the rank is at most $\min\{m,n\}$, we can distinguish some cases:

if $n<m$, the map can be surjective (when $k=n$), but not injective
if $n>m$, the map can be injective (when $k=m$), but not surjective
if $n=m$, the map is injective if and only if it is surjective (but it can be neither)