Endomorphism/Subspace condition/Matrices/Exercise

Let the vectors ${}u={\begin{pmatrix}6\\9\\-2\end{pmatrix}}$ and ${}v={\begin{pmatrix}4\\-4\\7\end{pmatrix}}$ in ${}\mathbb {R} ^{3}$ be given. We consider the linear subspace

${}U\subseteq \operatorname {Hom} _{K}{\left(\mathbb {R} ^{3},\mathbb {R} ^{3}\right)}\,$

consisting of all linear mappings ${}\varphi$ satisfying simultaneously the following conditions:

a) ${}\varphi (u)\in \langle {\begin{pmatrix}2\\8\\3\end{pmatrix}},{\begin{pmatrix}7\\0\\5\end{pmatrix}}\rangle$.

b) ${}\varphi (v)\in \langle {\begin{pmatrix}6\\-5\\4\end{pmatrix}}\rangle$.

1. Determine the dimension of ${}U$.
2. Describe the corresponding linear subspace of the space of matrices with respect to suitably chosen bases by linear equations.
3. Describe the corresponding linear subspace of the space of matrices with respect to suitably chosen bases by a basis.

Create a solution