Linear Transformation - proof isomorphism

Question

Hello I have the following question :

$W,V$ linear spaces with finite dimension above the same field.

$$T : V \rightarrow W$$ T is a linear transformation, let $U$ span of $V$ that impiles $V=U \oplus kerT$

$$S:U \rightarrow ImT$$

S is a linear transformation that is defined by $S(u)=T(u)$ for all $ u\in U $

Proof linear transformation $S$ isomorphism.

This is what I managed to get, I'm not so sure that I went in the right direction with this proof.

EDIT :

We know that : $$dimV=dimU+dimKerT$$ And as well that $$dimV=dimImT+dimKerT$$

Therefore we can conclude that $dimU=dimImT$

Any Idea how to process from this?

Thank you!

Answer 1

Your proof is excellent so far! You have $S:U \rightarrow \operatorname{Im}T$ is a linear transformation with $\dim(U) = \dim(\operatorname{Im}T)$.

In order to prove that $S$ is an isomorphism, it now is sufficient to show that $\ker(S) = \{0\}$. However, we know that this is true since $V = U \oplus \ker T$, which by definition implies that $U \cap \ker T = \{0\}$.