Base change/7-4/81 and 46/73/Point with coordinates -25/General/Exercise

We consider the families of vectors

${\mathfrak {v}}={\begin{pmatrix}7\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}4\\6\end{pmatrix}},\,{\begin{pmatrix}7\\3\end{pmatrix}}$

in ${}\mathbb {R} ^{2}$.

a) Show that ${}{\mathfrak {v}}$ and ${}{\mathfrak {u}}$ are both a basis of ${}\mathbb {R} ^{2}$.

b) Let ${}P\in \mathbb {R} ^{2}$ denote the point which has the coordinates ${}(-2,5)$ with respect to the basis ${}{\mathfrak {v}}$. What are the coordinates of this point with respect to the basis ${}{\mathfrak {u}}$?

c) Determine the transformation matrix which describes the change of bases

from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$.

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