Here is an example of change of variables in a linear way,
\begin{align*}
u &= 3x - 2y \\
v &= x + y
\end{align*}
In the language of linear algebra, we can rewrite is as:
\[
\begin{pmatrix}
u \\
v
\end{pmatrix}
=
\begin{pmatrix}
3 & -2 \\
1 & 1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
or,
\[
\begin{pmatrix}
u \\
v
\end{pmatrix}
=
x
\begin{pmatrix}
3 \\
1
\end{pmatrix}
+y
\begin{pmatrix}
-2 \\
1
\end{pmatrix}
\]
which creates a new coordinate system with $(3, 1)$ and
$(-2, 1)$ as the new axes.

## Saturday, April 5, 2014

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