Linear Transformations

Functions between vector spaces that preserve addition and scaling — and why every such function is represented by a matrix.

A 2 by 2 matrix is determined by where it sends the two basis vectors

[ 0.81 -0.59 ]

[ 0.59 0.81 ]

theta=36

The red and green arrows are the columns of the matrix. Once those are known, every point in the plane follows.

Definition

A linear transformation (or linear map) is a function $T: V \to W$ between vector spaces that preserves addition and scalar multiplication:

$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ (additivity)
$T(c\mathbf{u}) = c \cdot T(\mathbf{u})$ (homogeneity)

Equivalently: $T(a\mathbf{u} + b\mathbf{v}) = aT(\mathbf{u}) + bT(\mathbf{v})$ .

Every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be represented as multiplication by a matrix $A$ : $T(\mathbf{x}) = A\mathbf{x}$ .

Key properties

$T(\mathbf{0}) = \mathbf{0}$ always — a necessary (but not sufficient) condition for linearity
Linear transformations preserve straight lines, the origin, and parallelism
The composition of two linear transformations is itself linear
A linear transformation is fully determined by its effect on any basis of the domain

Common mistakes

Confusing linear with affine: translations, and any " $+c$ " shift, are not linear transformations — they fail $T(\mathbf{0})=\mathbf{0}$
Assuming linearity from one test point: checking $T(\mathbf{0})=\mathbf{0}$ alone isn't enough — both additivity and homogeneity must hold for every input

2D transformations

Rotation by $\theta$ : $\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}$
Scaling: $\begin{pmatrix}s_x & 0\\ 0 & s_y\end{pmatrix}$
Reflection across x-axis: $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$
Projection onto x-axis: $\begin{pmatrix}1&0\\0&0\end{pmatrix}$

Each is a linear transformation with its matrix representation.

Try it

Is the function $T(x, y) = (x + 1, y)$ (translation by 1 in the x direction) a linear transformation? Why or why not?

Solution

No. Additivity fails: $T(\mathbf{u}+\mathbf{v}) = (u_1+v_1+1, u_2+v_2)$ , but $T(\mathbf{u})+T(\mathbf{v}) = (u_1+1+v_1+1, u_2+v_2) = (u_1+v_1+2, u_2+v_2)$ . These differ.

Also: $T(\mathbf{0}) = (1, 0) \neq \mathbf{0}$ , which violates a necessary condition ( $T(\mathbf{0}) = \mathbf{0}$ for any linear transformation).

Translations are affine transformations, not linear. In homogeneous coordinates, $T(x,y) = (x+1,y)$ can be represented linearly as $(x,y,1) \mapsto (x+1,y,1)$ — extending the space by one dimension.

Related concepts

Algebra· Linear Algebra

MatricesRectangular arrays of numbers that encode systems of equations, transformations, and data — the central objects of linear algebra.

Algebra· Linear Algebra

Eigenvalues and EigenvectorsThe special directions a linear transformation merely scales — central to PCA, differential equations, and Google's PageRank.

Algebra· Linear Algebra

RankThe dimension of a matrix's column space — measuring how many linearly independent directions its transformation covers.

Algebra· Linear Algebra

OrthogonalityWhen vectors are perpendicular — dot product zero — and orthogonal projections, the foundation of least squares and QR decomposition.