Prove that $\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)$

Question

Prove that $\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)$ for every subspace $\mathbb{F}$ and every linear transformation $L$ of a vector space $V$ of a finite dimension.

Theorem: If $L:U\rightarrow V$ is a linear transformation and $\dim U=n$, then $\dim Ker L+\dim C(L^T)=n$.

$Ker L$ is the null space, $C(L^T)$ is the row space of $L$ and $n$ is is number of column vectors in $[L]$.

Question: How to use this theorem to prove the given statement?

Answer 1

Hint: $$L|_{F+\ker L}:F+\ker L\longrightarrow L(F)$$ is onto (why?) and now apply the first isomorphism theorem (the kernel is... because...).