Use Rank Nullity Theorem

Question

Let $A$ be a $3\times 4$ matrix and $B$ be a $4\times 3$ matrix. Show that the linear map $x\mapsto BAx$ from $\mathbb{R}^4$ to $\mathbb{R}^4$ cannot be one to one.

I am not sure how it is not one to one. I thought If $BA$ is a $4\times 4$ matrix its column and rows would equal and it would be one to one.

Answer 1

By the rank-nullity theorem, $\dim\ker A\geqslant1$. But $\ker A\subset\ker(BA)$. Therefore, $\dim\ker(BA)\geqslant1$.

Answer 2

Alternatively, assume suppose that the map is one-to-one. Then by dimension theorem $BA$ has full rank. So we have $$4=\text{rank}(BA) \leq \text{rank}A \leq 3$$ This absurdity proves the result!