Linear Transformations and Their Matrices

One-line statement: Every linear transformation between finite-dimensional vector spaces, once bases are chosen, is represented uniquely by a matrix, and composition of transformations corresponds to matrix multiplication. Key result in words: Linear maps and matrices are two faces of the same object; the matrix entries are the coordinates of the images of basis vectors. Where it applies: Linear algebra, differential equations, computer graphics, quantum mechanics, data science, control theory — anywhere structure-preserving maps appear. Who/when: Crystallized by Cayley, Sylvester, and Grassmann (1840s–1850s); the abstract framework matured with Peano (1888) and the 20th-century axiomatization of vector spaces.

A linear transformation $T : V \to W$ is a function between vector spaces that preserves the two fundamental operations: vector addition and scalar multiplication. Once we fix a basis for the domain $V$ and a basis for the codomain $W$ , the transformation is completely determined by a finite array of scalars — its matrix. This matters because it converts abstract, coordinate-free questions about maps (injectivity, surjectivity, eigenstructure) into concrete computations with numbers. The surprising fact is the depth of the correspondence: not only does every linear map between finite-dimensional spaces arise from a matrix, but the operations of composing maps, inverting them, and changing coordinates all translate into clean matrix algebra. The matrix is not the transformation — it is the transformation wearing the clothes of a particular pair of bases.

The Core Insight

A linear transformation is determined entirely by what it does to a basis. If ${v_{1}, \dots, v_{n}}$ spans the domain, then knowing $T (v_{1}), \dots, T (v_{n})$ pins down $T (v)$ for every $v$ , because $v$ is a unique linear combination of the basis vectors and $T$ respects combinations. Pack the coordinate vectors of $T (v_{1}), \dots, T (v_{n})$ as columns, and you have the matrix. Matrix columns are images of basis vectors.

Formal Statement

Linear Transformation

Let $V$ and $W$ be vector spaces over a field $F$ (typically $R$ or $C$ ). A function $T : V \to W$ is a linear transformation (or linear map) if for all $u, v \in V$ and all scalars $c \in F$ :

T (u + v) = T (u) + T (v)

⟳This is the additivity axiom; it is taken as a defining property, not derived. Apply

T

to a sum and require equality with the sum of images.

T (c u) = c T (u)

⟳Homogeneity axiom. Combined with [1], it gives

T (c_{1} u_{1} + c_{2} u_{2}) = c_{1} T (u_{1}) + c_{2} T (u_{2})

by applying [1] then [2].

Here $V$ is the domain, $W$ the codomain. The symbols $u, v$ denote vectors, $c$ a scalar. Both conditions together are equivalent to preservation of arbitrary linear combinations.

Matrix of a Linear Transformation

Let $T : V \to W$ be linear, with $dim V = n$ , $dim W = m$ . Choose an ordered basis $B = {v_{1}, \dots, v_{n}}$ for $V$ and $C = {w_{1}, \dots, w_{m}}$ for $W$ . Each image $T (v_{j})$ expands uniquely as

T (v_{j}) = i = 1 \sum m a_{ij} w_{i}, j = 1, \dots, n

⟳

T (v_{j}) \in W

, and

C

is a basis, so unique coordinates

a_{ij}

exist by the definition of basis (spanning + linear independence).

The scalars $a_{ij}$ form the matrix of $T$ with respect to $B$ and $C$ :

[T]_{C \leftarrow B} = (a_{ij}) \in F^{m \times n}

⟳Collect the coordinate column of

T (v_{j})

as the

j

-th column. The first index

i

is the row (output basis index), the second index

j

is the column (input basis index).

Fundamental Correspondence

Fix bases $B$ for $V$ and $C$ for $W$ . Let $[v]_{B}$ denote the coordinate column vector of $v$ . Then for every $v \in V$ :

[T (v)]_{C} = [T]_{C \leftarrow B} [v]_{B}

⟳Write

v = \sum_{j} x_{j} v_{j}

, apply linearity, substitute [3], and reindex. The matrix-times-vector product reproduces exactly the coordinate computation.

Moreover, the assignment $T \mapsto [T]_{C \leftarrow B}$ is a bijective linear isomorphism between $L (V, W)$ (the space of linear maps) and $F^{m \times n}$ . Composition corresponds to matrix multiplication:

[S \circ T]_{D \leftarrow B} = [S]_{D \leftarrow C} [T]_{C \leftarrow B}

⟳Apply [5] twice. The associativity of map composition forces the matrix product (and is the conceptual reason matrix multiplication is defined the way it is).

Why It Holds / Derivation

Proof of the Fundamental Correspondence, Equation 5

Step 1. Let

v \in V

. Since

B = {v_{1}, \dots, v_{n}}

is a basis, write

v = x_{1} v_{1} + x_{2} v_{2} + \dots + x_{n} v_{n},

so that

[v]_{B} = (x_{1}, \dots, x_{n})^{⊤}

is its coordinate vector. Step 2. Apply

T

and use linearity (equations [1] and [2] extended to combinations):

T (v) = T (j = 1 \sum n x_{j} v_{j}) = j = 1 \sum n x_{j} T (v_{j}) .

Step 3. Substitute the expansion [3],

T (v_{j}) = \sum_{i = 1}^{m} a_{ij} w_{i}

T (v) = j = 1 \sum n x_{j} i = 1 \sum m a_{ij} w_{i} = i = 1 \sum m (j = 1 \sum n a_{ij} x_{j}) w_{i} .

Step 4. Read off the

i

-th coordinate of

T (v)

in basis

C

[T (v)]_{C}^{(i)} = j = 1 \sum n a_{ij} x_{j} .

This is precisely the

i

-th entry of the matrix product

[T]_{C \leftarrow B} [v]_{B}

. Step 5. Hence

[T (v)]_{C} = [T]_{C \leftarrow B} [v]_{B}

as required.

■

Bijectivity: Given any matrix

A \in F^{m \times n}

, define

T

T (v_{j}) = \sum_{i} a_{ij} w_{i}

and extend linearly; this is well-defined because the

v_{j}

are a basis (a linear map is uniquely determined by, and freely specifiable on, basis vectors). The inverse maps

T

back to

A

, so the correspondence is a bijection. Linearity in

T

follows entry-wise.

Arthur Cayley | 1821–1895

The English mathematician Arthur Cayley introduced matrices as standalone algebraic objects in his 1858 "A Memoir on the Theory of Matrices," where he defined matrix addition, scalar multiplication, and — crucially — matrix multiplication as the operation that composes the linear substitutions the matrices represent. Cayley recognized that the strange-looking multiplication rule was forced by the requirement that matrices represent composed linear changes of variables. He also stated (verifying it for $2 \times 2$ and $3 \times 3$ cases) what is now the Cayley–Hamilton theorem. A barrister by profession for fourteen years, Cayley published nearly 1000 papers and helped found the abstract, structural style of modern algebra in which linear transformations and their matrices are inseparable.

Building Intuition

Think of a linear transformation as a way of re-stretching and re-shearing space while keeping the origin fixed and grid lines straight, parallel, and evenly spaced. The defining axioms [1] and [2] are exactly the conditions that guarantee this: lines map to lines, parallelograms to parallelograms, and the origin stays put.

Now picture the standard grid of $R^{2}$ with basis arrows $e_{1} = (1, 0)$ and $e_{2} = (0, 1)$ . To know everything $T$ does, you only need to know where these two arrows land — say $T (e_{1})$ and $T (e_{2})$ . Place those two landing vectors side by side as columns and you have the $2 \times 2$ matrix. To transform any point, decompose it along the grid (its coordinates) and follow the warped arrows the same distances. That's why a matrix-vector product is "a weighted recipe of the columns": the input coordinates are the weights, the columns are the transformed basis vectors.

The change-of-basis idea adds one twist: the same transformation looks like different matrices in different bases, the way the same physical rotation has different numerical descriptions in different coordinate systems. The transformation is the invariant truth; the matrix is the dialect.

A linear map of the unit square: basis vectors become matrix columns

Worked Examples

Rotation in the Plane

Given: Find the matrix of the counterclockwise rotation

T_{θ} : R^{2} \to R^{2}

by angle

θ = 30°

, using the standard basis, then apply it to

v = (2, 0)

. Step 1 — Image of $e_{1} = (1, 0)$ . Rotating

(1, 0)

θ

gives

(cos θ, sin θ)

. This is the first column. Step 2 — Image of $e_{2} = (0, 1)$ . Rotating

(0, 1)

θ

gives

(- sin θ, cos θ)

. This is the second column. Step 3 — Assemble the matrix.

[T_{θ}] = (cos θ sin θ - sin θ cos θ) = (\frac{3}{2} \frac{1}{2} - \frac{1}{2} \frac{3}{2}) .

Step 4 — Apply to $v = (2, 0)^{⊤}$ via equation [5].

[T_{θ}] (20) = (2 \cdot \frac{3}{2} + 0 2 \cdot \frac{1}{2} + 0) = (31) .

Answer:

T_{30°} (2, 0) = (3, 1) \approx (1.732, 1.000)

The length is preserved:

(3)^{2} + 1^{2} = 4 = 2

, as expected for a rotation (units: same as input, here dimensionless plane coordinates).

Differentiation as a Matrix on Polynomials

Given: Let

V = P_{3}

, the space of polynomials of degree

\leq 3

, with basis

B = {1, x, x^{2}, x^{3}}

. Let

W = P_{2}

with basis

C = {1, x, x^{2}}

. Find the matrix of the derivative operator

D : P_{3} \to P_{2}

D (p) = p^{'}

, and apply it to

p (x) = 5 + 4 x + 3 x^{2} + 2 x^{3}

. Step 1 — Differentiate each basis vector.

D (1) = 0

D (x) = 1

D (x^{2}) = 2 x

D (x^{3}) = 3 x^{2}

. Step 2 — Coordinate columns in $C$ .

$D (1) = 0 \to (0, 0, 0)^{⊤}$
$D (x) = 1 \to (1, 0, 0)^{⊤}$
$D (x^{2}) = 2 x \to (0, 2, 0)^{⊤}$
$D (x^{3}) = 3 x^{2} \to (0, 0, 3)^{⊤}$

Step 3 — Assemble the $3 \times 4$ matrix (equation [4]).

[D]_{C \leftarrow B} = 000100020003 .

Step 4 — Apply via [5].

[p]_{B} = (5, 4, 3, 2)^{⊤}

0001000200035432 = 466 .

Step 5 — Translate back. Coordinates

(4, 6, 6)

C

mean

4 + 6 x + 6 x^{2}

. Answer:

D (p) = 4 + 6 x + 6 x^{2}

Verify directly:

\frac{d}{d x} (5 + 4 x + 3 x^{2} + 2 x^{3}) = 4 + 6 x + 6 x^{2}

. ✓

Common Misconceptions

A function that "looks linear" need not be a linear transformation. The map

f (x) = m x + b

from school algebra is affine, not linear, whenever

b \neq = 0

: it violates

f (0) = 0

, which every linear map must satisfy (set

c = 0

in [2]). Likewise,

T (x) = x^{2}

T (x) = ∣ x ∣

, and

T (x) = x + c

are all nonlinear. Every linear transformation sends $0 \mapsto 0$ .

A second trap: the matrix of $T$ depends on the chosen bases. The phrase "the matrix of $T$ " is meaningless without specifying $B$ and $C$ . Two correct-looking matrices can represent the very same map; they are related by change-of-basis (similarity, when domain = codomain). See Change of Basis.

Consider $T : R^{2} \to R$ defined by $T (x_{1}, x_{2}) = x_{1} x_{2}$ . Test additivity with $u = (1, 0)$ , $v = (0, 1)$ :

T (u + v) = T (1, 1) = 1, T (u) + T (v) = 0 + 0 = 0.

Since

1 \neq = 0

, equation [1] fails —

T

is not linear, despite being a "simple" polynomial of the coordinates. No matrix can represent it. Multilinear (here bilinear) does not mean linear; the product of coordinates is the classic non-example.

Connections

Linear transformations and their matrices sit at the hub of linear algebra. The column space of $[T]$ is the image of $T$ and the null space is its kernel, linking directly to the Rank-Nullity Theorem, which states $dim (ker T) + dim (im T) = dim V$ . The question "when is $[T]$ invertible?" connects to the Determinant (nonzero iff invertible, for square matrices) and to Systems of Linear Equations (a unique solution exists iff $T$ is bijective). When domain equals codomain, the search for bases in which $[T]$ is diagonal leads to Eigenvalues and Eigenvectors and the Spectral Theorem. The dependence of $[T]$ on the chosen basis is governed by Change of Basis and similarity transformations $P^{- 1} A P$ . Finally, the space $L (V, W)$ of all linear maps is itself a vector space isomorphic to $F^{m \times n}$ , tying back to Vector Spaces and Dual Spaces.

Real-World Applications

Computer graphics & games. Rotations, scalings, shears, and (in homogeneous coordinates) translations and perspective projections are all matrix transformations; the GPU pipeline multiplies vertex vectors by model–view–projection matrices millions of times per frame.
Quantum mechanics. Observables are linear (Hermitian) operators; their matrices in a chosen basis encode measurable spectra. The evolution operator $e^{- i H t /ℏ}$ is a unitary linear transformation. See Spectral Theorem.
Principal Component Analysis (data science). PCA finds the linear transformation (an orthogonal change of basis) that diagonalizes the covariance matrix, compressing high-dimensional data along directions of maximal variance.
Control theory & robotics. State-space models $\dot{x} = A x + B u$ use matrices to describe how linear systems evolve; controllability and stability are read off matrix structure (eigenvalues of $A$ ).
Markov chains. A transition matrix is the linear transformation advancing a probability distribution one step; long-run behavior comes from its dominant eigenvector.
Differential equations. Systems $x^{'} = A x$ are solved by exponentiating the matrix; diagonalization decouples the equations.
Cryptography & coding theory. Linear (Hamming, Reed–Muller) codes encode messages as $c = G m$ via a generator matrix $G$ , a linear transformation over a finite field.

Historical Development

1683 / 1693 — Seki Takakazu in Japan and Leibniz in Europe independently study determinants while solving linear systems, an early hint of arrays governing linear relations.
1750 — Cramer publishes Cramer's rule, expressing solutions of linear systems through determinants.
1844 — Hermann Grassmann's Ausdehnungslehre introduces an abstract, coordinate-free theory of linear combinations and dimension — far ahead of its time and poorly understood by contemporaries.
1850 — James Joseph Sylvester coins the term "matrix" (Latin for "womb"), viewing it as an array from which minors and determinants are born.
1858 — Arthur Cayley's Memoir on the Theory of Matrices defines matrix algebra abstractly and establishes that matrix multiplication encodes composition of linear substitutions.
1878 — Georg Frobenius systematizes rank, canonical forms, and the theory of linear substitutions, proving the Cayley–Hamilton theorem in general.
1888 — Giuseppe Peano gives the first modern axiomatic definition of a vector space (his "linear systems"), placing linear transformations on rigorous footing.
1918 — Hermann Weyl, in Space, Time, Matter, and the broader development of functional analysis extend linear operators to infinite-dimensional spaces.
1920s–1930s — The matrix formulation of quantum mechanics (Heisenberg, Born, Jordan, 1925) makes linear operators and their matrices central to physics, cementing the topic's universal importance.

Subtleties & Limits

The clean matrix correspondence is a finite-dimensional luxury. In infinite-dimensional spaces (function spaces, Hilbert spaces), linear operators need not be represented by finite matrices, and new phenomena appear: unbounded operators (like differentiation), operators with continuous spectrum, and the failure of every operator to have eigenvalues. The invariant subspace problem — does every bounded linear operator on an infinite-dimensional separable Hilbert space have a nontrivial closed invariant subspace? — remains open in general (Enflo and Read settled the Banach-space case negatively in the 1980s, but the Hilbert-space case is unresolved). Even in finite dimensions, classification up to similarity over a non-algebraically-closed field is intricate, requiring rational canonical form; and numerically, computing eigenstructure of nearly-defective matrices is ill-conditioned, a practical limit on how faithfully a matrix reveals its transformation's behavior.