Tool for calculating a change of basis matrix based on a homothety or rotation in a vector space and coordinate change calculations.
Change of Basis Matrix - dCode
Tag(s) : Matrix
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Change of Basis Matrix
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Change of Basis Equations Calculator
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Tool for calculating a change of basis matrix based on a homothety or rotation in a vector space and coordinate change calculations.
Answers to Questions
How to calculate change of basis equations?
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} $
How to calculate a rotation matrix?
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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