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.
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?