Change of Basis Matrix

From a transformation matrix $ P $ (also called base change of basis matrix), any vector $ v $ then becomes the vector $ v' $ in the new base by the computation (dot / multiplication">matrix product) $$ v' = P.v $$

Example: $ \begin{pmatrix} v_1' \\ v_2' \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} . \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} $

From a rotation angle $ \alpha $ (trigonometric direction) and an axis, the rotation matrix is written as (rotation around the axis $ z $) $$ \begin {pmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha \cos \alpha & 0 \\ 0 & 0 & 1 \ \end{pmatrix} $$

From 2 vectors (the original and the destination one), it is possible to generate an equation system to solve to find the values of $ \alpha $ and the axis.

From the value of the scaling factor $ k $ (homothety assumed to be uniform throughout the vector space of size $ n $), the passing matrix is given by the formula $$ k.I_n $$ (with $ I_n $ the identity matrix).