Linear transformations with restricted range

Question

Let $W$ be a subspace of a vector space $V$ over a finite field $F$ and let $L(V,W)$ denote the set of all linear transformations from $V$ into $W$. Let $f$ be an element in $L(V,W)$ with $f(V)\subseteq f(W)$. How to prove the following problems.

1. If $g\in L(V,W)$ such that $f(V)\neq g(V)\subseteq g(W)$ and $\dim(g(V)) = \dim(f(V))$, then there exists $h\in L(V,W)$ such that $f(V)\neq h(V)\subseteq h(W), \dim(h(V)) = \dim(f(V))$ and $g = g\circ h\circ g$.

2. If $g\in L(V,W)$ such that $g(V)\subseteq g(W), \ker(g)\neq \ker(f)$ and $\dim(g(V)) = \dim(f(V))$, then there exists $h\in L(V,W)$ such that $h(V)\subseteq h(W), \ker(h)\neq \ker(f)$ and $g = g\circ h\circ g$.

Thank you so much.