Let $T:V \to U$ be an one-to-one linear transformation and $S:U \to W$ an onto linear transformation.

Question

Let $T:V \to U$ be an one-to-one linear transformation and $S:U \to W$ an onto linear transformation. Is $SoT:V \to W$ isomorphic?

Is this condition always true or is there any good counterexamples to show otherwise?

Answer 1

In general, there is no reason why $S\circ T$ should either be injective or surjective.

In fact, it's not too hard to come up with an example where $T$ is injective, $S$ is surjective, and $S\circ T$ is the zero map: let $T$ be the map from $\mathbb{R}$ to $\mathbb{R}^2$ defined by $x\mapsto (x,0)^T$, and $S$ the map $\mathbb{R}^2\to\mathbb{R}$ defined by $(x,y)^T\mapsto y$.