In this app, you can choose components for the matrix $M in M^2$ and the vector $u in R^2$, along with an angle θ of the counterclockwise rotation of the coordinate system. The app then applies the transformation of coordinates form the coordinate system $B = {e_1, e_2}$, to $B' = {e_1', e_2'}$, where $e_1^{\mathrm{\prime }}e\_1\text{'}$ , $e_2^{\mathrm{\prime }}e\_2\text{'}$ are vectors rotated by θ counterclockwise from $e_{1}e\_1$ , and $e_{2}e\_2$. The expressions for $u,M,v,e_1^{\mathrm{\prime }}u,\; M,\; v,\; e\_1\text{'}$ and $e_2^{\mathrm{\prime }}e\_2\text{'}$ using the basis set $BB$ are shown. The expressions for $u^{\mathrm{\prime }},M^{\mathrm{\prime }}u\text{'},\; M\text{'}$ and $v^{\mathrm{\prime }}v\text{'}$ are also shown using the basis set $B^{\mathrm{\prime }}B\text{'}$. The first plot the app draws shows the vector $uu$ in blue, the vector $v=Muv\; =\; Mu$ in red, the original coordinate system in black, and the vectors of the new coordinate system in gray. The second plot the app draws shows the vector $u^{\mathrm{\prime }}u\text{'}$ in blue, $v^{\mathrm{\prime }}v\text{'}$ in red and the new coordinate system in black.