If $V$ is finite-dimensional and exist $\beta$ basis of $V$ such that $T(\beta)$ is a basis for $W$, then $T$ is a isomorphism?

Question

Let $V$ and $W$ vector space over $F$ and $T : V \rightarrow W$ lineal. The statement is false, but I can't find a counterexample.

Answer 1

Consider $V=\Bbb{R}^2$ and $W=\Bbb{R}^1$, and let $\pi:V\to W$, $\pi(x,y)=x$. Let $\beta=\{(1,0),(1,1)\}$, then $\pi(\beta)=\{(1)\}$, which is basis of $W$.