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 $U$?