User:IssaRice/Linear algebra/Change of basis example in two dimensions

This example comes from this video.

We are working in ${\mathbf{R}}^2$, the plane.

Jennifer's basis vectors: ${\overrightarrow{\mathbf{b}}}_1:=\left[\begin{array}{c}2\\ 1\end{array}\right]$ and ${\overrightarrow{\mathbf{b}}}_2:=\left[\begin{array}{c}-1\\ 1\end{array}\right]$

To Jennifer, ${\overrightarrow{\mathbf{b}}}_1$ looks like $\left[\begin{array}{c}1\\ 0\end{array}\right]$ and ${\overrightarrow{\mathbf{b}}}_2$ looks like $\left[\begin{array}{c}0\\ 1\end{array}\right]$.