I'd like to get some help in a type of problem I'm a little bit unfamiliar with, which is finding basis for function vector spaces
Let $V$ be the space of endomorphisms of $\mathbb{R}^{3}$, then we consider $$S:=\{f \in V \ : T \subseteq \text{Ker}(f)\}$$ with $$T:=\{(x_{1},x_{2},x_{3}) \in \mathbb{R}^{3} : x_{1}=x_{2}\}.$$
I'm asked to find $\dim_{\mathbb{R}}{S}$ and $\dim_{\mathbb{R}}{\text{Im}(f)}$ for any $f \in S$.
One of the things I tried is attempting to find the basis for the matrices associated with the functions in $S$ and make an argument using the isomorphism between matrices and linear transformations. To that end, I used the standard basis of $\mathbb{R^3}$, then applied an arbitrary function $f \in S$ to each of my basis vectors, which would send $(0,0,1)$ to $(0,0,0)$ and the other two basis vectors to an unknown pair of vectors $(a_{1,1},a_{1,2},a_{1,3})$ and $(a_{2,1},a_{1,2},a_{2,3})$, which gives me the matrix $$ \begin{bmatrix} a_{1,1} & a_{2,1} & 0 \\ a_{1,2} & a_{2,2} & 0 \\ a_{1,3} & a_{2,3} & 0 \\ \end{bmatrix} $$
I'm pretty sure I'm mistaken somewhere, since this matrix doesn't seem to send every vector in $T$ to $(0,0,0)$, but I can't find the mistake. If anyone could tell me where I went wrong and also a nudge in the right direction to solve this problem I'd greatly appreciate it. Cheers!
You are right that $(0,0,1)$ should be sent to $(0,0,0)$, but you have not used all the information about $T$ that you have.
In particular, if you consider what $f \in S$ would do to $(1,1,0)$, you see that you should have $a_{11} + a_{21} = 0$, $a_{12}+a_{22}=0$, and $a_{13}+a_{23}=0$ as well.