Linear Transformations — Mathematics Knowledge System

Overview

A linear transformation (or linear map) is a function between two vector spaces that respects vector addition and scalar multiplication. Every linear transformation between finite-dimensional spaces can be represented by a matrix, and applying the transformation to a vector is exactly matrix-vector multiplication — this correspondence is what makes matrices useful for more than just bookkeeping systems of equations.

Intuition

A linear transformation is a function that treats 'straight lines through the origin' consistently: it never bends space, only stretches, rotates, reflects, shears, or projects it — the origin always stays fixed, and grid lines (if you picture a coordinate grid warping) remain straight, evenly spaced, and parallel. If a transformation would curve a straight line or move the origin, it isn't linear.

Interactive Graph

Formal Definition

Definition

A function T: V → W between vector spaces V and W (over the same field) is a linear transformation if it preserves addition and scalar multiplication for all vectors u, v and scalars c:

T: V \to W

T maps vectors in V to vectors in W

T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})

T respects vector addition

Additivity

T(c\mathbf{v}) = c\, T(\mathbf{v})

T respects scalar multiplication

Homogeneity

T(\mathbf{0}_V) = \mathbf{0}_W

Consequence: every linear transformation sends the zero vector to the zero vector

Notation

Notation	Meaning
$T: V \to W$	A linear transformation from V to W
$\ker(T) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}$	The kernel (null space) — vectors mapped to zero
$\operatorname{im}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\}$	The image (range) of T
$[T]_{\mathcal{B}}$	The matrix representing T with respect to basis 𝓑

Properties

Matrix representation

\text{Every } T: \mathbb{R}^n \to \mathbb{R}^m \text{ linear has a unique matrix } A \text{ with } T(\mathbf{x}) = A\mathbf{x}

Example: The columns of A are T(e₁), …, T(eₙ), the images of the standard basis vectors

Kernel is a subspace

\ker(T) \text{ is a subspace of } V

Rank-nullity theorem

\dim(\ker T) + \dim(\operatorname{im} T) = \dim(V)

Composition corresponds to matrix multiplication

(S \circ T)(\mathbf{v}) = [S][T]\mathbf{v}

Condition: For linear T:V→W and S:W→U represented by matrices

Applications

Rotations, scalings, reflections, and projections in 2D/3D graphics are all linear transformations, each representable as a single matrix applied to every vertex.

Worked Examples

Check additivity: T((x₁,y₁)+(x₂,y₂)) should equal T(x₁,y₁)+T(x₂,y₂).
$T(x_1+x_2, y_1+y_2) = (2(x_1+x_2), (x_1+x_2)+(y_1+y_2))$
This equals (2x₁+2x₂, x₁+y₁+x₂+y₂) = T(x₁,y₁) + T(x₂,y₂), so additivity holds; homogeneity T(cx,cy)=(2cx,cx+cy)=c·T(x,y) also holds.
$T(c\mathbf{v}) = c\,T(\mathbf{v})\ \checkmark$

Answer: Yes, T is linear, with matrix [[2,0],[1,1]].

Practice Problems

Difficulty 4/10

Is f(x) = x + 1 (as a map ℝ→ℝ) a linear transformation? Justify.

Difficulty 5/10

T: ℝ³ → ℝ² is linear with dim(ker T) = 1. By the rank-nullity theorem, what is dim(im T)?

1230

Difficulty 5/10

In computer graphics, scaling by 2 in x and 3 in y is the linear transformation with matrix [[2,0],[0,3]]. Where does the unit square with corner (1,1) map, and by what factor does area change?

Common Mistakes

Common Mistake

Calling any function that 'looks like a line' (e.g. f(x)=2x+3) a linear transformation.

In linear algebra, 'linear' has a strict meaning: T(0)=0 is required. f(x)=2x+3 fails since f(0)=3≠0 — such functions are called affine, not linear.

Common Mistake

Building the matrix of a transformation by plugging in arbitrary vectors instead of the standard basis vectors.

The columns of [T] are exactly T(e₁), T(e₂), …, T(eₙ) — the images of the standard basis. Using any other vectors won't directly give you the matrix's columns.

Quiz

Summary

A linear transformation T: V → W preserves vector addition and scalar multiplication: T(u+v)=T(u)+T(v), T(cv)=cT(v).
Every linear T between finite-dimensional spaces is represented by a matrix; the matrix's columns are the images of the standard basis vectors.
The kernel (null space) and image (range) are subspaces; rank-nullity relates their dimensions: dim(ker T) + dim(im T) = dim(V).
Composing linear transformations corresponds exactly to multiplying their matrices.
Rotations, reflections, scalings, and neural-network layers are all everyday examples of linear transformations.

References

WebsiteWikipedia — Linear map
BookAxler, S. Linear Algebra Done Right, 3rd ed. Ch. 3.