I have an assignment, where we're trying to prove certain properties and behaviors between vector spaces, subspaces and their relationships with transformations.
Let V,W be vector spaces with a finite number of vectors as their basis, and let H $\subseteq V$, be a subspace of $V$.
$T:V \to W $ is linear
We know that $T(H)\subseteq W$ is a subspace of $W$
We define the function $S$ to be the following:
$S:H \to W$ , $S(h) = T(h)$
Then could I assume that $Im(S) =T(H)$ is true? I'm fairly certain it is, though I couldn't find formal proof anywhere.
It's fairly obvious and straighforward ($Im(S)=S(H)$), but here's a formal proof:
Let $x \in Im(S)$. Then there exists $h\in H$ such that $S(h)=x$. Then, $T(h)=S(h)=x$ and $x \in T(H)$. Thus, $Im(S) \subseteq T(H)$.
Conversely, let $x \in T(H)$, then there exists $h \in H$ such that $T(h)=x$. Then, $S(h)=T(h)=x$ and $x \in Im(S)$. Thus, $T(H) \subseteq Im(S)$.
We conclude $Im(S)=T(H)$.