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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) $$ v' = P.v $$

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.

How to calculate an homothety matrix?

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).

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Thanks to your feedback and relevant comments, dCode has developed the best 'Change of Basis Matrix' tool, so feel free to write! Thank you!

Thanks to your feedback and relevant comments, dCode has developed the best 'Change of Basis Matrix' tool, so feel free to write! Thank you!