Let $V$ and $W$ be finite dimensional vector spaces over a field $F$. Let $T:V\to W$ be a linear transformation. Suppose that $T$ is one-to-one. Show that there is a linear transformation $L:W\to V$ such that $LT=1_V$.
The second question is Let $T:V\to W$ and $L:W\to V$ be a linear transformation. Show:
- $T$ is injective if $LT=1_V$ and
- $T$ is surjective if $TL=1_W$
I know that if 1 and 2 are true, that it is a isomophism, no idea how to prove it though.