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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

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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)
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The highest the rank of an $m \times n$ matrix can be is $\min\{m,n\}$ (full rank).

If the rank of a matrix is the highest it can be, then it is injective if $m\geq n$ and surjective if $m \leq n$. If a matrix does not have full rank, it is neither injective nor surjective.

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