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

We consider the families of vectors

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

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

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

b) Let ${}P\in \mathbb {R} ^{3}$ denote the point that has the coordinates ${}(2,5,4)$ 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 that describes the change of basis from ${}{\mathfrak {v}}$ to ${}{\mathfrak {u}}$.

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