We consider in
the
standard basis,
-
![{\displaystyle {}{\mathfrak {u}}={\begin{pmatrix}1\\0\end{pmatrix}},\,{\begin{pmatrix}0\\1\end{pmatrix}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3bee1dcb9586bfe06056274a18b24cc0428a89)
and the basis
-
![{\displaystyle {}{\mathfrak {v}}={\begin{pmatrix}1\\2\end{pmatrix}},\,{\begin{pmatrix}-2\\3\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5da4d3fccaa31c7a94cec47a4a37e9d5ccc37ff)
The basis vectors of
can be expressed directly with the standard basis, namely
-
Therefore, we get immediately
-
![{\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}={\begin{pmatrix}1&-2\\2&3\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56752c9184b6968c8043f32417e8eff761a23f4)
For example, the vector which has with respect to
the
coordinates
, 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}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b069dd235603eeb58b8459bf90fe9cf0352fe644)
with respect to the standard basis
. The transformation matrix
is more difficult to compute: We have to write the standard vectors as
linear combinations
of
and
.
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}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c51b564f7927ac8f563310964f1924130b3fd55)
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}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72e3999f67057322774731fe65b8324dc6929b75)
Hence,
-
![{\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}={\begin{pmatrix}{\frac {3}{7}}&{\frac {2}{7}}\\-{\frac {2}{7}}&{\frac {1}{7}}\end{pmatrix}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a25e80c6901f9b1c9204e3426639ba7b572d5a0c)