Linear Transformations

A linear transformation is a function T: ℝⁿ → ℝᵐ between vector spaces that preserves two essential operations: vector addition and scalar multiplication. Formally, for any vectors u and v and any scalar c, we require T(u + v) = T(u) + T(v) and T(cu) = cT(u). These two conditions together guarantee that the transformation respects the underlying structure of vector spaces — straight lines map to straight lines (or the origin), parallelograms map to parallelograms, and the origin always maps to the origin. Every linear transformation from ℝⁿ to ℝᵐ can be represented by an m × n matrix A, so that T(x) = Ax for every vector x in ℝⁿ. This matrix representation is one of the most powerful ideas in linear algebra: it reduces the study of all possible linear transformations to the study of matrix multiplication. Geometric operations such as rotations, reflections, scaling, shearing, and projections can all be encoded as matrices and applied to vectors with a single matrix-vector product. When two or more linear transformations are applied in sequence, the result is called a composition of transformations. Remarkably, the composition of two linear transformations is itself a linear transformation, and its matrix is simply the product of the individual matrices. This connection between function composition and matrix multiplication is the reason matrix multiplication is defined the way it is, and it makes linear algebra an indispensable tool in computer graphics, physics simulations, machine learning, and data science.

Key takeaways
  • Checking Linearity
  • Finding the Standard Matrix
  • Common Geometric Transformations in ℝ²
  • Composition of Transformations
  • Kernel and Image

Checking Linearity

To verify that a function T is a linear transformation, check both linearity conditions. First, T(u + v) = T(u) + T(v) (additivity). Second, T(cu) = cT(u) (homogeneity). A useful shortcut: both conditions combine into T(au + bv) = aT(u) + bT(v) for all scalars a, b. If T(0) ≠ 0, the function cannot be linear, since T(0) = T(0·v) = 0·T(v) = 0 must always hold.

Finding the Standard Matrix

Every linear transformation T: ℝⁿ → ℝᵐ has a unique standard matrix A whose columns are the images of the standard basis vectors. If e₁, e₂, …, eₙ are the standard basis vectors of ℝⁿ, then A = [T(e₁) | T(e₂) | … | T(eₙ)]. For example, if T: ℝ² → ℝ² and T(e₁) = (2, 1)ᵀ and T(e₂) = (−1, 3)ᵀ, then A = [[2, −1],[1, 3]]. Once you have A, you compute T(x) for any x simply by multiplying: T(x) = Ax.

Common Geometric Transformations in ℝ²

Several fundamental geometric operations have well-known standard matrices. A counterclockwise rotation by angle θ is given by R(θ) = [[cos θ, −sin θ],[sin θ, cos θ]]. Reflection across the x-axis uses [[1, 0],[0, −1]], and reflection across y = x uses [[0, 1],[1, 0]]. A horizontal shear by factor k has matrix [[1, k],[0, 1]], and a uniform scaling by factor s uses [[s, 0],[0, s]]. Memorizing these templates lets you build complex transformations quickly.

Composition of Transformations

If T₁: ℝⁿ → ℝᵐ has matrix A and T₂: ℝᵐ → ℝᵖ has matrix B, then the composition (T₂ ∘ T₁)(x) = T₂(T₁(x)) has standard matrix BA. Note the order: the matrix of the transformation applied first appears on the right. Applying T₁ then T₂ gives BA·x, not AB·x. Matrix multiplication is generally not commutative, so the order matters enormously. For instance, rotating then reflecting usually gives a different result than reflecting then rotating.

Kernel and Image

Two subspaces capture the essential behavior of a linear transformation. The kernel (or null space) of T is ker(T) = {x ∈ ℝⁿ : T(x) = 0}, the set of all vectors sent to zero. The image (or range) of T is im(T) = {T(x) : x ∈ ℝⁿ}, the set of all possible outputs. The Rank-Nullity Theorem states: dim(ker T) + dim(im T) = n, where n is the dimension of the domain. This powerful equation relates the 'information lost' by T to the 'information preserved.'

The Key Equation

T(x)=Ax,A=[T(e1)T(e2)⋯T(en)]T(\mathbf{x}) = A\mathbf{x}, \quad A = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \end{bmatrix}
  • T — Linear transformation (none): A function from ℝⁿ to ℝᵐ satisfying additivity and homogeneity
  • A — Standard matrix (none): The unique m × n matrix whose columns are T(e₁), T(e₂), …, T(eₙ)
  • \mathbf{x} — Input vector (none): Any vector in the domain ℝⁿ
  • \mathbf{e}_i — Standard basis vector (none): The vector with a 1 in position i and 0s elsewhere

Worked Example

Problem. Let T₁: ℝ² → ℝ² be a counterclockwise rotation by 90°, and let T₂: ℝ² → ℝ² be a reflection across the x-axis. (a) Write the standard matrices A and B for T₁ and T₂. (b) Find the matrix of the composition T₂ ∘ T₁ (rotate first, then reflect). (c) Apply this composition to the vector v = (3, 1)ᵀ.

BA=[0−1−10],(T2∘T1)(31)=(−1−3)BA = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}, \quad (T_2 \circ T_1)\begin{pmatrix}3\\1\end{pmatrix} = \begin{pmatrix}-1\\-3\end{pmatrix}

Answer. The composition matrix is BA = [[0, −1],[−1, 0]], and it maps v = (3, 1)ᵀ to (−1, −3)ᵀ.

Figure
Diagram
Effect of a 90° CCW Rotation on ℝ²xyv=(3,1)T(v)=(-1,3)O90°xye₂e₁T(e₂)T(e₁)Left: vector v and its 90° rotation T(v). Right: basis vectors e₁,e₂ and their images under R(90°).
Origins · 1 minArthur Cayley and how this was discovered
AC
Arthur Cayley
1821–1895 · British

Arthur Cayley was a 19th-century British mathematician who founded the modern theory of matrices. In 1858, he published 'A Memoir on the Theory of Matrices,' in which he introduced the algebra of matrices — including matrix addition, multiplication, and the concept of the inverse matrix. Cayley recognized that sets of linear equations could be represented compactly as matrix equations Ax = b, and he proved that every square matrix satisfies its own characteristic polynomial (the Cayley-Hamilton theorem). His framework established the foundation upon which linear transformations are studied today. Despite working as a lawyer for 14 years to support himself financially, Cayley still managed to publish nearly 300 mathematical papers during that period — an astonishing output for anyone, let alone a practicing attorney.

  • Matrix Multiplication — The composition of two linear transformations corresponds exactly to the product of their standard matrices, making matrix multiplication the core computational tool for composing transformations.
  • Vector Spaces — Linear transformations are defined as maps between vector spaces; understanding the vector space axioms is essential for grasping why linearity is such a natural and powerful condition.
  • Determinant — The determinant of a transformation's matrix measures how the transformation scales areas or volumes, and tells you whether the transformation is invertible.
  • Eigenvalues and Eigenvectors — Eigenvectors are special vectors whose direction is unchanged by a linear transformation; eigenvalues measure the factor by which those vectors are scaled.