In the following app, you can choose the components of the matrix M and the vector u along with an angle theta of the counterclockwise rotation of the coordinate system. The app plots the vector u in blue, the vector v=Mu in red, the original coordinate system in black, and the vectors of the new coordinate system in gray. The app also plots the vectors u' in blue, v'=M'u' in red, and the new coordinate system in black. At the bottom of the right hand side, you will find the expressions for u', M', and v' using the basis set B'.