checking of injectiveness of the function

Question

Let $A\in M_{n\times n}(\mathbb{R})$ be of rank $m.$ Then the map $\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ given by $v\rightarrow Av$ is not injective.

Answer 1

The matrix $A$ should be in $M_{m\times n}(\mathbb{R})$ in order for $v\mapsto Av$ to be a linear map $\mathbb{R}^n\to\mathbb{R}^m$.

If $n>m$, then the map cannot be injective because of the rank-nullity theorem.

If $n=m$, the map is injective.

It is not possible that $n<m$, because in this case the matrix cannot have rank $m$.