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

Transformation	Matrix
Reflection in the x-axis	$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$(10​0−1​)
Reflection in the y-axis	$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$(−10​01​)
Reflection in $y = x$y=x	$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$(01​10​)
Reflection in $y = -x$y=−x	$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$(0−1​−10​)
Rotation by $\theta$θ anticlockwise	$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$(cosθsinθ​−sinθcosθ​)
Enlargement, scale factor $k$k	$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$(k0​0k​)
Stretch in $x$x-direction, factor $a$a	$\begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix}$(a0​01​)
Shear (x-direction), factor $k$k	$\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$(10​k1​)

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Worked Examples

Example 1: Find the matrix representing rotation by 90° anticlockwise.

$\theta = 90°$θ=90°: $\cos 90° = 0$cos90°=0, $\sin 90° = 1$sin90°=1.

$M = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$M=(01​−10​)

Check: $\begin{pmatrix} 1 \\ 0 \end{pmatrix} \mapsto \begin{pmatrix} 0 \\ 1 \end{pmatrix}$(10​)↦(01​) (90° anticlockwise ✓).

Example 2: Find the image of the point $(3, 1)$(3,1) under reflection in $y = x$y=x.

$$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$$(01​10​)(31​)=(13​)

The image is $(1, 3)$(1,3).

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Combining Transformations

If transformation $S$S is represented by matrix $A$A and transformation $T$T by matrix $B$B, then:

• $T$T followed by $S$S is represented by $AB$AB

The order matters: the transformation applied first is the right matrix.

Worked Example 3: Find the matrix representing a reflection in the x-axis followed by a rotation of 90° anticlockwise.

Reflection in x-axis: $A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$A=(10​0−1​). Rotation 90° anticlockwise: $B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$B=(01​−10​).

"Reflection followed by rotation" means reflection first, then rotation: $BA$BA.