Linear Transformations in 2D

Matrices can represent linear transformations — geometric operations such as reflections, rotations, enlargements, and shears. This lesson covers how to find the matrix of a 2D transformation and how to combine transformations using matrix multiplication.

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What Is a Linear Transformation?

A transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$T:R2→R2 is linear if:

1. $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$T(u+v)=T(u)+T(v)
2. $T(k\mathbf{u}) = kT(\mathbf{u})$T(ku)=kT(u)

Every linear transformation of $\mathbb{R}^2$R2 can be represented by a 2×2 matrix $M$M:

$T\begin{pmatrix} x \\ y \end{pmatrix} = M\begin{pmatrix} x \\ y \end{pmatrix}$T(xy​)=M(xy​)

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Finding the Matrix

The matrix $M$M has columns that are the images of the standard basis vectors $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$e1​=(10​) and $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$e2​=(01​):

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

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Standard 2D Transformation Matrices