Linear Transformations and Their Matrices

⚡ Key Takeaways

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.

💡 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.

DefinitionLinear Transformation

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

1

\quad T(u + v) = T(u) + T(v)

Derivation hint: 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.

2

\quad T(c\,u) = c\,T(u)

Derivation hint: 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.

DefinitionMatrix of a Linear Transformation

Let $T: V \to W$ be linear, with $\dim V = n$ , $\dim W = m$ . Choose an ordered basis $\mathcal{B} = \{v_1, \dots, v_n\}$ for $V$ and $\mathcal{C} = \{w_1, \dots, w_m\}$ for $W$ . Each image $T(v_j)$ expands uniquely as

3

\quad T(v_j) = \sum_{i=1}^{m} a_{ij}\, w_i, \qquad j = 1, \dots, n

Derivation hint: $T(v_j) \in W$ , and $\mathcal{C}$ 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 $\mathcal{B}$ and $\mathcal{C}$ :

4

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

Derivation hint: 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).

TheoremFundamental Correspondence

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

5

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

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

6

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

Derivation hint: 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).

ProofProof of the Fundamental Correspondence, Equation 5

Step 1. Let $v \in V$ . Since $\mathcal{B} = \{v_1, \dots, v_n\}$ is a basis, write

v = x_1 v_1 + x_2 v_2 + \cdots + x_n v_n,

so that $[v]_\mathcal{B} = (x_1, \dots, x_n)^\top$ is its coordinate vector.

Step 2. Apply $T$ and use linearity (equations [1] and [2] extended to combinations):

T(v) = T\!\left(\sum_{j=1}^n x_j v_j\right) = \sum_{j=1}^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) = \sum_{j=1}^n x_j \sum_{i=1}^m a_{ij} w_i = \sum_{i=1}^m \left(\sum_{j=1}^n a_{ij} x_j\right) w_i.

Step 4. Read off the $i$ -th coordinate of $T(v)$ in basis $\mathcal{C}$ :

\big[T(v)\big]_\mathcal{C}^{(i)} = \sum_{j=1}^n a_{ij} x_j.

This is precisely the $i$ -th entry of the matrix product $[T]_{\mathcal{C}\leftarrow\mathcal{B}}\,[v]_\mathcal{B}$ .

Step 5. Hence $[T(v)]_\mathcal{C} = [T]_{\mathcal{C}\leftarrow\mathcal{B}}\,[v]_\mathcal{B}$ as required. $\blacksquare$

Bijectivity: Given any matrix $A \in \mathbb{F}^{m\times n}$ , define $T$ by $T(v_j) = \sum_i a_{ij} w_i$ 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.

∎

mathematician · 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.

🧠 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 $\mathbb{R}^2$ 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.

Figure

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

ExampleRotation in the Plane

Given: Find the matrix of the counterclockwise rotation $T_\theta: \mathbb{R}^2 \to \mathbb{R}^2$ by angle $\theta = 30°$ , using the standard basis, then apply it to $v = (2, 0)$ .

Step 1 — Image of $e_1 = (1,0)$ . Rotating $(1,0)$ by $\theta$ gives $(\cos\theta, \sin\theta)$ . This is the first column.

Step 2 — Image of $e_2 = (0,1)$ . Rotating $(0,1)$ by $\theta$ gives $(-\sin\theta, \cos\theta)$ . This is the second column.

Step 3 — Assemble the matrix.

[T_\theta] = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} = \begin{pmatrix} \tfrac{\sqrt3}{2} & -\tfrac12 \\[2pt] \tfrac12 & \tfrac{\sqrt3}{2} \end{pmatrix}.

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

[T_\theta] \begin{pmatrix}2\\0\end{pmatrix} = \begin{pmatrix} 2\cdot\tfrac{\sqrt3}{2} + 0 \\ 2\cdot\tfrac12 + 0 \end{pmatrix} = \begin{pmatrix}\sqrt3 \\ 1\end{pmatrix}.

Answer:

\boxed{T_{30°}(2,0) = (\sqrt3,\ 1) \approx (1.732,\ 1.000)}

The length is preserved: $\sqrt{(\sqrt3)^2 + 1^2} = \sqrt{4} = 2$ , as expected for a rotation (units: same as input, here dimensionless plane coordinates).

ExampleDifferentiation as a Matrix on Polynomials

Given: Let $V = P_3$ , the space of polynomials of degree $\le 3$ , with basis $\mathcal{B} = \{1, x, x^2, x^3\}$ . Let $W = P_2$ with basis $\mathcal{C} = \{1, x, x^2\}$ . Find the matrix of the derivative operator $D: P_3 \to P_2$ , $D(p) = p'$ , and apply it to $p(x) = 5 + 4x + 3x^2 + 2x^3$ .

Step 1 — Differentiate each basis vector. $D(1) = 0$ , $D(x) = 1$ , $D(x^2) = 2x$ , $D(x^3) = 3x^2$ .

Step 2 — Coordinate columns in $\mathcal{C}$ .

$D(1) = 0 \to (0,0,0)^\top$
$D(x) = 1 \to (1,0,0)^\top$
$D(x^2) = 2x \to (0,2,0)^\top$
$D(x^3) = 3x^2 \to (0,0,3)^\top$

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

[D]_{\mathcal{C}\leftarrow\mathcal{B}} = \begin{pmatrix} 0&1&0&0 \\ 0&0&2&0 \\ 0&0&0&3 \end{pmatrix}.

Step 4 — Apply via [5]. $[p]_\mathcal{B} = (5,4,3,2)^\top$ :

\begin{pmatrix} 0&1&0&0 \\ 0&0&2&0 \\ 0&0&0&3 \end{pmatrix}\!\begin{pmatrix}5\\4\\3\\2\end{pmatrix} = \begin{pmatrix}4\\6\\6\end{pmatrix}.

Step 5 — Translate back. Coordinates $(4,6,6)$ in $\mathcal{C}$ mean $4 + 6x + 6x^2$ .

Answer:

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

Verify directly: $\tfrac{d}{dx}(5 + 4x + 3x^2 + 2x^3) = 4 + 6x + 6x^2$ . ✓

⚠️ Common mistake

A function that "looks linear" need not be a linear transformation. The map $f(x) = mx + b$ 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 $T(\mathbf{x}) = \mathbf{x} + \mathbf{c}$ are all nonlinear. Every linear transformation sends $\mathbf{0} \mapsto \mathbf{0}$ . A second trap: the matrix of $T$ depends on the chosen bases. The phrase "the matrix of $T$ " is meaningless without specifying $\mathcal{B}$ and $\mathcal{C}$ . 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.

Counterexample

Consider $T: \mathbb{R}^2 \to \mathbb{R}$ 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, \qquad 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.

connection

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 $\dim(\ker T) + \dim(\operatorname{im} T) = \dim V$ . 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 $P^{-1}AP$ . Finally, the space $\mathcal{L}(V,W)$ of all linear maps is itself a vector space isomorphic to $\mathbb{F}^{m\times n}$ , tying back to Vector Spaces and Dual Spaces.

🔬 Where this appears

⏳ History

- 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.

open-problem

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.

Linear Transformations and Their Matrices

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also