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Change of Basis Matrix

Tool for calculating a change of basis matrix based on a homothety or rotation in a vector space and coordinate change calculations.

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Change of Basis Matrix -

Tag(s) : Matrix

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# Change of Basis Matrix

## Rotation Matrix Calculator

### From rotation data in 3D

 Rotation axis Axis X Axis Y Axis Z

## Homothety Matrix Calculator

### 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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