Change of variables and change of coordinate system

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.