Linear Transformations in 3D

This lesson extends linear transformations to three dimensions. We use 3×3 matrices to represent rotations, reflections, and other transformations in 3D space. The principles are the same as in 2D, but the geometry is richer.

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3D Transformation Matrices

A linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$T:R3→R3 is represented by a 3×3 matrix $M$M. The columns of $M$M are the images of the standard basis vectors:

$M = \begin{pmatrix} | & | & | \\ T(\mathbf{e}_1) & T(\mathbf{e}_2) & T(\mathbf{e}_3) \\ | & | & | \end{pmatrix}$M=​∣T(e1​)∣​∣T(e2​)∣​∣T(e3​)∣​​

where $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$e1​=​100​​, $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$e2​=​010​​, $\mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$e3​=​001​​.

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Standard 3D Transformations

Reflections