If two linear transformations are one to one, is the product of them also one to one?

Question

Example:

Let $T : V \to W$ and $S : W \to U$ be two one to one linear transformations.

Would $S \circ T$ also be one to one?

Answer 1

Yes, and we need not even consider linear transformations. This result holds for any functions.

If $S\circ T(y)=S\circ T(x)$ then by injectivity of $S$ we have $T(y)=T(x)$. By injectivity of $T$ we then have $x=y$, hence $S\circ T$ is injective.