Base change/Transformation matrix/R^4 to R^3/Exercise

We consider the families of vectors

${\mathfrak {u}}={\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\1\\1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {v}}={\begin{pmatrix}1\\0\\1\end{pmatrix}},\,{\begin{pmatrix}0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}0\\0\\1\end{pmatrix}}$

in ${}\mathbb {R} ^{4}$ and in ${}\mathbb {R} ^{3}$. We denote the standard bases by ${}{\mathfrak {e}}_{4}$ and ${}{\mathfrak {e}}_{3}$. We consider the linear mapping

$f\colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{3}$

given by the matrix

${}A={\begin{pmatrix}-2&3&2&3\\-3&5&0&1\\-1&2&-2&-2\end{pmatrix}}\,$

with respect to the standard bases. Determine the describing matrices of ${}f$ with respect to the bases

a) ${}{\mathfrak {e}}_{4}$ and ${}{\mathfrak {v}}$,

b) ${}{\mathfrak {u}}$ and ${}{\mathfrak {e}}_{3}$,

c) ${}{\mathfrak {u}}$ and ${}{\mathfrak {v}}$.

Create a solution