Linear mapping/K^3 to K^2/125/411/On hyperplane given by 234/Several basises/Exercise

We consider the linear map

$\varphi \colon K^{3}\longrightarrow K^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}1&2&5\\4&1&1\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}.$

Let ${}U\subseteq K^{3}$ be the subspace of ${}K^{3}$, defined by the linear equation ${}2x+3y+4z=0$, and let ${}\psi$ be the restriction of ${}\varphi$ on ${}U$. On ${}U$, there are given vectors of the form

$u=(0,1,a),\,v=(1,0,b){\text{ and }}w=(1,c,0).$

Compute the "change of basis" matrix between the bases

${\mathfrak {b}}_{1}=v,w,\,{\mathfrak {b}}_{2}=u,w{\text{ and }}{\mathfrak {b}}_{3}=u,v$

of ${}U$, and the transformation matrix of ${}\psi$ with respect to these three bases

(and the standard basis of ${}K^{2}$).

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