Let $V$ be a vector space over a field $F$. Let $S,T\colon V\to F$ be linear transformations. Assume that $N(T)\subseteq N(S)$. Show that there exists $a\in F$ such that $S=aT$.
$N(T)$ and $N(S)$ are the null spaces.
I've messed around with this problem for a couple hours but I didn't really get anywhere. I thought about showing that $S(v)/T(v)$ is a constant for all $v$ or showing that $S$ and $T$ are linearly dependent in the vector space $L(V,F)$ of all linear transformations from $V$ to $F$.
Can anyone outline a solution?
Either $\ker S=V$, in which case just let $a=0$, or $S\neq 0$, in which case $\ker T\subseteq\ker S$ gives $\operatorname{codim}\ker T\geqslant 1$ and hence $\ker T=\ker S$ (since $\operatorname{codim}\ker T\leqslant 1$ from $T\colon V\to F$). Pick $v\in V$ with $v\notin\ker T$ so $Tv\neq 0$ and let $a=\dfrac{Sv}{Tv}$. Since $\operatorname{codim}\ker T=1$ we have every $v'\in V$ satisfies $v'-\lambda v\in\ker T$ for some $\lambda\in F$ and hence $$Sv'=\bar{S}(v'+\ker T)=\lambda\bar{S}(v+\ker T)=\lambda aTv=aTv'$$ where $\bar{S}\colon V/\ker T\to F$ is the induced map from $\ker T\subseteq\ker S$.