# Base change/123/471/025 and 024/661/35-2/Point with coordinates 254/General/Exercise

We consider the families of vectors

${\displaystyle {\mathfrak {v}}={\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}4\\7\\1\end{pmatrix}},\,{\begin{pmatrix}0\\2\\5\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}0\\2\\4\end{pmatrix}},\,{\begin{pmatrix}6\\6\\1\end{pmatrix}},\,{\begin{pmatrix}3\\5\\-2\end{pmatrix}}}$

in ${\displaystyle {}\mathbb {R} ^{3}}$.

a) Show that ${\displaystyle {}{\mathfrak {v}}}$ and ${\displaystyle {}{\mathfrak {u}}}$ are both a basis of ${\displaystyle {}\mathbb {R} ^{3}}$.

b) Let ${\displaystyle {}P\in \mathbb {R} ^{3}}$ denote the point which has the coordinates ${\displaystyle {}(2,5,4)}$ with respect to the basis ${\displaystyle {}{\mathfrak {v}}}$. What are the coordinates of this point with respect to the basis ${\displaystyle {}{\mathfrak {u}}}$?

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

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