How do you find the matrix of a linear transformation?

Apply the transformation to each standard basis vector e₁, e₂, …, eₙ and arrange the results as columns. For abstract vector spaces, apply T to each basis vector of V and express the results as coordinate vectors in the basis of W. The resulting matrix satisfies T(x) = Ax for all x.

Why does every linear transformation have a matrix?

Linearity forces T(x) = x₁T(e₁) + ⋯ + xₙT(eₙ) for any vector x. This is exactly the matrix-vector product Ax where column j of A is T(eⱼ). The correspondence is one-to-one: different matrices define different transformations, and vice versa.

How does matrix multiplication relate to composition?

If T has matrix A and S has matrix B, then the composition S ∘ T has matrix BA. The left-to-right order of matrix multiplication matches the outer-to-inner order of composition. This is why matrix multiplication is defined by the row-times-column rule.

What does the matrix of the differentiation operator look like?

For differentiation T: P₂ → P₁ with monomial bases, the matrix is [[0, 1, 0], [0, 0, 2]]. Column j records the coefficients of T applied to the j-th basis polynomial. The rank is 2 (differentiation maps onto all of P₁) and the kernel is the constant polynomials.

Does the matrix of a transformation depend on the basis?

Yes. Different bases produce different matrices for the same transformation. The standard matrix uses the standard basis for both domain and codomain. For abstract spaces like polynomial spaces, every basis choice gives a different but equally valid matrix representation.

Matric Representation of Linear Transformation

Every Linear Map from Rⁿ to Rᵐ Is Matrix Multiplication

Constructing the Standard Matrix

Reading the Matrix

Matrices for Abstract Vector Spaces

Worked Example: Differentiation

Composition Corresponds to Matrix Multiplication

The Identity and the Inverse

The Matrix Encodes Everything

Summary: The Transformation ↔ Matrix Dictionary

Encoding a Transformation as a Matrix

Every linear transformation between finite-dimensional spaces can be represented by a matrix, and every matrix defines a linear transformation. The columns of the matrix are the images of the basis vectors — this single recipe converts an abstract function into a concrete array of numbers from which every property of the transformation can be extracted.

Every Linear Map from Rⁿ to Rᵐ Is Matrix Multiplication

T: \mathbb{R}^n \to \mathbb{R}^m

is linear, there exists a unique

m \times n

matrix

A

such that

T(\mathbf{x}) = A\mathbf{x} \quad \text{for every } \mathbf{x} \in \mathbb{R}^n

This is not an optional representation — it is forced by linearity. Any vector

\mathbf{x} = x_1\mathbf{e}_1 + \cdots + x_n\mathbf{e}_n

maps to

T(\mathbf{x}) = x_1T(\mathbf{e}_1) + \cdots + x_nT(\mathbf{e}_n)

, and this is exactly the matrix-vector product

A\mathbf{x}

with

A = [T(\mathbf{e}_1) \; T(\mathbf{e}_2) \; \cdots \; T(\mathbf{e}_n)]

.

The converse is equally immediate: every

m \times n

matrix

A

defines a linear transformation

\mathbf{x} \mapsto A\mathbf{x}

. The correspondence is one-to-one — different matrices define different transformations, and different transformations produce different matrices. Linear maps

\mathbb{R}^n \to \mathbb{R}^m

and

m \times n

matrices are the same objects viewed from two perspectives.

Constructing the Standard Matrix

The recipe is direct: apply

T

to each standard basis vector

\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n

and arrange the results as columns:

A = \begin{pmatrix} | & | & & | \\ T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \\ | & | & & | \end{pmatrix}

Worked Example

Let

T: \mathbb{R}^3 \to \mathbb{R}^2

be defined by

T(x, y, z) = (2x - y + 3z, \; 4x + 5z)

T(\mathbf{e}_1) = T(1, 0, 0) = (2, 4)

T(\mathbf{e}_2) = T(0, 1, 0) = (-1, 0)

T(\mathbf{e}_3) = T(0, 0, 1) = (3, 5)

A = \begin{pmatrix} 2 & -1 & 3 \\ 4 & 0 & 5 \end{pmatrix}

Verification:

A\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2x - y + 3z \\ 4x + 5z \end{pmatrix} = T(x, y, z)

Reading the Matrix

Every piece of information about a linear transformation is encoded in its matrix.

Column

j

tells you where

\mathbf{e}_j

goes. If the first column of a

2 \times 2

matrix is

(3, -1)^T

, then

(1, 0)

maps to

(3, -1)

. The matrix is a complete lookup table: the image of any vector is computed by multiplication.

The size

m \times n

records the dimensions of codomain (

m

rows) and domain (

n

columns). A

3 \times 2

matrix represents a map from

\mathbb{R}^2

\mathbb{R}^3

— it embeds a plane into three-dimensional space. A

2 \times 3

matrix represents a map from

\mathbb{R}^3

\mathbb{R}^2

— it compresses three dimensions down to two.

A square matrix (

m = n

) represents a transformation from a space to itself — a linear operator. Only operators can have eigenvalues, determinants, and traces.

Shape of A (m × n)	Transformation type	Geometric action
m > n (more rows than columns)	T: ℝⁿ → ℝᵐ, embedding	embeds a smaller-dimensional space into a larger one (e.g. 3 × 2 lifts ℝ² into ℝ³)
m = n (square)	T: ℝⁿ → ℝⁿ, linear operator	maps a space to itself — only operators carry determinant, eigenvalues, and trace
m < n (more columns than rows)	T: ℝⁿ → ℝᵐ, compression	compresses a larger-dimensional space into a smaller one (e.g. 2 × 3 flattens ℝ³ into ℝ²)

Matrices for Abstract Vector Spaces

For a linear transformation

T: V \to W

between abstract vector spaces, the matrix depends on a choice of basis for both

V

and

W

.

Fix a basis

\mathcal{B} = \{\mathbf{v}_1, \dots, \mathbf{v}_n\}

for

V

and a basis

\mathcal{C} = \{\mathbf{w}_1, \dots, \mathbf{w}_m\}

for

W

. Column

j

of the matrix

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

is the

\mathcal{C}

-coordinate vector of

T(\mathbf{v}_j)

— the scalars needed to express

T(\mathbf{v}_j)

as a linear combination of

\mathbf{w}_1, \dots, \mathbf{w}_m

.

Different bases give different matrices for the same transformation. The standard matrix for maps between

\mathbb{R}^n

and

\mathbb{R}^m

is the special case where both bases are standard. For abstract spaces like polynomial or function spaces, there is no "standard" basis in the same sense — every basis choice produces a different but equally valid matrix representation.

Worked Example: Differentiation

Let

T: \mathcal{P}_2 \to \mathcal{P}_1

be defined by

T(p) = p'

(differentiation). Choose the monomial basis

\{1, x, x^2\}

for

\mathcal{P}_2

and

\{1, x\}

for

\mathcal{P}_1

T(1) = 0 = 0 \cdot 1 + 0 \cdot x

, so column

1

(0, 0)^T

T(x) = 1 = 1 \cdot 1 + 0 \cdot x

, so column

2

(1, 0)^T

T(x^2) = 2x = 0 \cdot 1 + 2 \cdot x

, so column

3

(0, 2)^T

[T] = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}

The

2 \times 3

shape reflects

\dim(\mathcal{P}_1) = 2

rows and

\dim(\mathcal{P}_2) = 3

columns. The rank is

2

— differentiation maps

\mathcal{P}_2

onto all of

\mathcal{P}_1

. The null space is one-dimensional, spanned by the constant polynomial

1

— the only polynomials whose derivative is zero.

Composition Corresponds to Matrix Multiplication

T: U \to V

has matrix

A

(relative to appropriate bases) and

S: V \to W

has matrix

B

, then the composition

S \circ T: U \to W

has matrix

BA

.

The order matches the composition:

S

acts after

T

, and

B

multiplies from the left. This is why matrix multiplication is defined as it is — the row-times-column rule encodes function composition.

Associativity of matrix multiplication

(BC)A = B(CA)

mirrors associativity of composition

(R \circ S) \circ T = R \circ (S \circ T)

. Non-commutativity of multiplication

AB \neq BA

mirrors non-commutativity of composition

S \circ T \neq T \circ S

.

The identity transformation

I: V \to V

has the identity matrix

I_n

in any basis, and

I_n A = AI_n = A

— the identity matrix is the multiplicative identity precisely because the identity transformation is the compositional identity.

The Identity and the Inverse

The identity transformation

I: V \to V

sends every vector to itself. In any basis, its matrix is

I_n

.

If

T

has matrix

A

and

T

is invertible, then

T^{-1}

has matrix

A^{-1}

. The transformation

T

is invertible if and only if

A

is invertible — the geometric and algebraic conditions coincide exactly. Composing

T

with

T^{-1}

gives the identity, and multiplying

A

A^{-1}

gives

I

. The determinant test (

\det(A) \neq 0

) simultaneously answers whether the transformation is bijective and whether the matrix has an inverse.

The Matrix Encodes Everything

Once a linear transformation is represented by a matrix, every property of the transformation becomes a matrix computation.

The rank of

A

equals the dimension of the image of

T

. The nullity equals the dimension of the kernel. The determinant (for square matrices) tells whether

T

is invertible and how it scales volumes. The eigenvalues reveal the scaling factors along invariant directions. The trace equals the sum of the eigenvalues. The singular values measure the maximum stretching in each orthogonal direction.

This is why matrices dominate computational linear algebra. Abstract transformations are conceptually powerful, but matrices are what computers operate on. The matrix representation converts every question about a linear map into a question about an array of numbers — and arrays of numbers are what algorithms are designed to handle.

Matrix invariant	What it reveals about T	Requires
rank(A)	dim of the image of T (how many output directions T can reach)	any A
nullity(A)	dim of the kernel of T (how many input directions T collapses to zero)	any A
det(A)	invertibility (det ≠ 0); volume scaling factor \|det(A)\|; orientation by sign	square A only
eigenvalues	scaling factors along the invariant directions (eigenvectors)	square A only
tr(A)	sum of the eigenvalues	square A only
singular values	maximum stretching along each orthogonal input direction	any A

Summary: The Transformation ↔ Matrix Dictionary

The recipe and examples above all serve a single end: each linear transformation between finite-dimensional spaces corresponds to a unique matrix, and every concept on the transformation side has a matching object on the matrix side. The table below collects the dictionary in one place — what to look at on the matrix when you want to know something about the transformation, and vice versa. It is a recap of the whole page and a reference for moving between the two languages.

Transformation-side concept	Matrix-side counterpart
Linear T: ℝⁿ → ℝᵐ	unique m × n matrix A
T(eⱼ) — image of j-th basis vector	j-th column of A
dim(domain) = n	number of columns of A
dim(codomain) = m	number of rows of A
Linear operator (V = W)	square matrix
Identity transformation I	identity matrix Iₙ
Composition S ∘ T	matrix product BA
Inverse T⁻¹	inverse matrix A⁻¹
T is invertible (bijective)	A is invertible (det(A) ≠ 0 for square A)
Image of T	column space of A
Kernel of T	null space of A
Isomorphism between V and W	invertible square matrix relating the two coordinate systems