[Math] Kernel decomposition of vector spaces

Question

Let $T:V\longrightarrow W$ be a linear map between vector spaces over the same field. I know that
$$V/\operatorname{Ker}(T)\cong\operatorname{Im}(T)$$
I want to deduce a relation of the form
$$V=\operatorname{Ker}(T)\oplus V'$$
where $V'$ is a subspace of $V$, isomorphic to the quotient $V/\operatorname{Ker}(T)$.
Is it possible? If yes, how can I prove it? If not, how can it be corrected?

Answer 1

Choose a basis for $\text{Ker}(T)$ and extend to a basis of $V$. This works even for infinite dimensional spaces as long as you accept the Axiom of Choice. The difference between the extended basis and the original basis generates a space $V'$ which satisfies $V = \text{Ker}(T) \oplus V'$. The projection $V \to V'$ induces an isomorphism $V/\text{Ker}(T) \cong V'$ by the usual isomorphism theorem.