Suppose $W$ is finite-dimensional and $T_1,T_2 \in \mathcal L(V,W).$ Prove that $\text{null } T_1 \subseteq \text{null }T_2$ if and only if there exists $S \in \mathcal L(W,W)$ such that $T_2 = ST_1$.

First let's consider the case where $V$ is finite-dimensional, before generalizing to $V$ being possibly infinite dimensional.

Suppose $\text{null }T_1 \subseteq \text{null }T_2$. Let $w_1,\dots,w_r$ form a basis for $\text{range }T_2$ then extend to a basis $w_1,\dots,w_m$ of $W$. Let $v_1,\dots,v_r$ be vectors in $V$ such that $T_2v_j = w_j$ then extend to a basis $v_1,\dots,v_n$ of $V$.
Notice $v_1,\dots,v_r$ is independent since if it were dependent, it would implythat $w_1,\dots,w_r$ is dependent which it isn't since it's a basis. (this can be made rigorous with the linear dependence lemma)

Define $S(T_1v_j) = T_2v_j = w_j$ when $j \le r$ and $Sw_j = 0$ when $j > r$. This definition makes sense because $T_1v_1,\dots,T_1v_r$ is independent (todo: rigor).
Any $v \in V$ can be written $v = a_1v_1+\dots+a_nv_n$. By linearity and $ST_1v_j = T_2v_j$ we get

Therefor $T_2 = ST_1$ over $\text{span}(v_1,\dots,v_n) = V$.

TODO: Handle the case where $V$ is infinite-dimensional, and make rigorous

First time I approached this I wasted a bunch of time trying to be uber-rigorous from the start. when I should have solved it quickly via a half-rigorous argument, then made it rigorous after.
An example: $\text{null }T_1 \subseteq \text{null }T_2$ implying that $Tv_1,\dots,Tv_r$ were independent, intuitively clear to me so I initially skipped the proof.

Rigorizing things that are intuitively clear breaks my thought process when I'm looking for a solution, from now on i'll come up with a solution, then prove it.

Also be lazy about proving things, when I introduced $v_1,\dots,v_r$ I was tempted to immediately prove it's independent but I held myself off.

This took me so long! More then 2h In total. Here's what I did wrong:

• Tried to be overly rigorous from the start
• Stayed obsessed on this one problem for way too long, switch it up when stuck!