Change of basis/R^2/Standard and 12,-23/Example

We consider in ${\displaystyle {}\mathbb {R} ^{2}}$ the standard basis,

${\displaystyle {}{\mathfrak {u}}={\begin{pmatrix}1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\1\end{pmatrix}}\,,}$

and the basis

${\displaystyle {}{\mathfrak {v}}={\begin{pmatrix}1\\2\end{pmatrix}},\,{\begin{pmatrix}-2\\3\end{pmatrix}}\,.}$

The basis vectors of ${\displaystyle {}{\mathfrak {v}}}$ can be expressed directly with the standard basis, namely

${\displaystyle v_{1}={\begin{pmatrix}1\\2\end{pmatrix}}=1{\begin{pmatrix}1\\0\end{pmatrix}}+2{\begin{pmatrix}0\\1\end{pmatrix}}\,\,{\text{ and }}\,\,v_{2}={\begin{pmatrix}-2\\3\end{pmatrix}}=-2{\begin{pmatrix}1\\0\end{pmatrix}}+3{\begin{pmatrix}0\\1\end{pmatrix}}.}$

Therefore, we get immediately

${\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}={\begin{pmatrix}1&-2\\2&3\end{pmatrix}}\,.}$

For example, the vector that has the coordinates ${\displaystyle {}(4,-3)}$ with respect to ${\displaystyle {}{\mathfrak {v}}}$, has the coordinates

${\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}{\begin{pmatrix}4\\-3\end{pmatrix}}={\begin{pmatrix}1&-2\\2&3\end{pmatrix}}{\begin{pmatrix}4\\-3\end{pmatrix}}={\begin{pmatrix}10\\-1\end{pmatrix}}\,}$

with respect to the standard basis ${\displaystyle {}{\mathfrak {u}}}$. The transformation matrix ${\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}}$ is more difficult to compute. We have to write the standard vectors as linear combinations of ${\displaystyle {}v_{1}}$ and ${\displaystyle {}v_{2}}$. A direct computation (solving two linear systems) yields

${\displaystyle {}{\begin{pmatrix}1\\0\end{pmatrix}}={\frac {3}{7}}{\begin{pmatrix}1\\2\end{pmatrix}}-{\frac {2}{7}}{\begin{pmatrix}-2\\3\end{pmatrix}}\,}$

and

${\displaystyle {}{\begin{pmatrix}0\\1\end{pmatrix}}={\frac {2}{7}}{\begin{pmatrix}1\\2\end{pmatrix}}+{\frac {1}{7}}{\begin{pmatrix}-2\\3\end{pmatrix}}\,.}$

Hence,

${\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}={\begin{pmatrix}{\frac {3}{7}}&{\frac {2}{7}}\\-{\frac {2}{7}}&{\frac {1}{7}}\end{pmatrix}}\,.}$