0
$\begingroup$

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 $U$?

Improve this question
$\endgroup$
  • $\begingroup$ $f$ occurs twice in different roles. $\endgroup$ – Henno Brandsma Jun 29 '18 at 9:21
  • $\begingroup$ I have edited . $\endgroup$ – fcoulomb Jun 29 '18 at 9:22
2
$\begingroup$

Hint:

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

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.