Find basis from matrix representation of linear transformation

Question

If I have a vector space like $U=\{f\in End(\mathbb{R}^3)$ such that $... \}$ (a certain condition) and I know that the matrix representation of $f$ (with respect to two basis of $\mathbb{R}^3$) is $$\begin{pmatrix} a & c & p \\ b & d & q\\ 0 & 0 & r \end{pmatrix}$$ with $a,b,c,p,q,r$ free variables.

Then can I find a basis of $U$?

Answer 1

Hint:

$$\begin{pmatrix} a & c & p \\ b & d & q\\ 0 & 0 & r \end{pmatrix}= a\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}+b\begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}+c\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}+\cdots$$