Linear mapping/Kernel and fiber/3/Exercise

Let

$\varphi \colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{2},{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}\longmapsto {\begin{pmatrix}6x-5y-3z+8w\\x+5y+4z-2w\end{pmatrix}}.$

Determine, for the set

${}E={\left\{Q\in \mathbb {R} ^{4}\mid \varphi (Q)={\begin{pmatrix}-3\\-7\end{pmatrix}}\right\}}\,,$

a description using a starting point and a translating space.

Create a solution