Matrix Representation of Transformations

Every Linear Map from Rⁿ to Rᵐ Is Matrix Multiplication

If

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

is linear, there exists a unique

m \times n

matrix

A

such that

Linear Map as Matrix Multiplication

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

Learn more about this formula: Linear Map as Matrix Multiplication →

Browse all linear algebra formulas →

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.

Matrix Representation Notation

Notation

Matrix Representation Notation

The same square brackets that turn a vector into a column turn a map into a matrix — and once a map is inside them, the subscripts have to say which bases were used, because the array changes when they do. All of these marks are catalogued among the linear algebra symbols.

The brackets themselves, the script-vs-roman basis letters and the backward-arrow subscript come from basis notation, where they wrap a vector;

T: V \to W

and the map letters come from image and kernel notation;

A\mathbf{x}

juxtaposition and the

m \times n

shape from matrix operation notation.

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

the matrix of T, from the B basis to the C basis

The bracket device of coordinate notation with a map inside instead of a vector: the result is the array that does what

T

does, once both ends have been given coordinates. The subscript is not decoration — Matrices for Abstract Vector Spaces below shows the same

T

producing different arrays under different bases.

CasesWhen both spaces carry the standard basis the subscript is dropped and the bare

[T]

or

A

is the standard matrix of Constructing the Standard Matrix below — the only case where a map has one uncontested array. An operator mapping a space to itself uses one basis twice, written

[T]_{\mathcal{B}}

rather than

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

.

Also written

A_T

,

M(T)

and

\mathcal{M}(T)

all appear for the same object; some texts drop the brackets entirely once a basis is fixed and simply reuse

T

for both the map and its matrix — convenient, and the source of the confusion below.

Do not confuseThe map itself.

T

is basis-free;

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

is one of infinitely many arrays representing it. Texts that write

T

for both are relying on a fixed basis the reader is expected to remember, and the distinction reappears the moment a change of basis does.

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

coordinates of the image equal the matrix times the coordinates

The equation the subscripts were designed for: adjacent basis letters touch and cancel,

\mathcal{C} \leftarrow \mathcal{B}

against

\mathcal{B}

, leaving

\mathcal{C}

on both sides. It is the same domino reading change-of-basis notation uses for

P_{\mathcal{C} \leftarrow \mathcal{B}}

, which is why both marks point their arrows the same way.

CasesChaining is the payoff:

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

— Composition Corresponds to Matrix Multiplication below — where the inner labels cancel in sequence and the surviving outer pair names the composite.

Do not confuseA left-to-right reading. The rightmost factor acts first, so the arrow chain runs backwards relative to the page:

\mathcal{B}

enters on the right and

\mathcal{D}

leaves on the left, matching function composition rather than reading order.

[T] = \big[\;[T(\mathbf{b}_1)]_{\mathcal{C}} \;\cdots\; [T(\mathbf{b}_n)]_{\mathcal{C}}\;\big]

column j is the image of the j-th basis vector, in coordinates

The construction rule written as a mark: each column is one basis vector's image, expressed in the target basis — Constructing the Standard Matrix below builds it this way. The outer brackets here are matrix-assembly, not coordinates; the inner ones are coordinates.

CasesWith the standard basis the inner brackets collapse, since

[\mathbf{v}]_{\mathcal{E}} = \mathbf{v}

, and the rule shortens to "columns are

T(\mathbf{e}_j)

" — the form most first courses state, and a special case rather than the definition.

Do not confuseRows. Reading the images across rows instead of down columns transposes the matrix and breaks

T(\mathbf{x}) = A\mathbf{x}

; the column convention is fixed by the fact that the matrix multiplies a column vector on its right.

All linear algebra symbols →↑ Linear Transformations

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:

Standard Matrix

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

Learn more about this formula: Standard Matrix →

Browse all linear algebra formulas →

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)

.

The basis vectors, and where they land

The images of the two basis vectors are the two columns of the matrix — that is the entire construction. Because a linear map is determined by what it does to a basis, knowing those two arrows is knowing the map, and everything else on the grid follows from them. Move the basis images and watch the matrix update on the linear transformation explorer.

This is why the standard matrix is built by feeding the basis vectors through the map rather than by solving anything.

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

to

\mathbb{R}^3

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

2 \times 3

matrix represents a map from

\mathbb{R}^3

to

\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

. In compact form:

Matrix Representation Abstract Bases

[T]_{\mathcal{C} \leftarrow \mathcal{B}} = \bigl[\,[T(\mathbf{v}_1)]_{\mathcal{C}} \;\; \cdots \;\; [T(\mathbf{v}_n)]_{\mathcal{C}}\,\bigr]

Learn more about this formula: Matrix Representation Abstract Bases →

Browse all linear algebra formulas →

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

is

(0, 0)^T

.

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

, so column

2

is

(1, 0)^T

.

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

, so column

3

is

(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

If

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.

Two maps applied in each order

Above, the order makes no difference. Below, it plainly does: the same two transformations applied the other way round leave the grid somewhere else. Matrix multiplication is non-commutative for this reason and no other — doing things in a different order genuinely ends up somewhere different. Swap the order yourself on the 2D composition explorer.

Composition of maps and multiplication of matrices are therefore the same operation described in two vocabularies.

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

by

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.

Read the entries below in both directions. Each line is a fact about transformations and a fact about matrices at the same time, and which one is the definition depends only on where you started. The grouping separates the correspondence itself from what it does to operations and to subspaces — three kinds of translation rather than one long glossary.

Download

PNGRaster · 2×PDFPrint quality

Transformations · dictionary

Transformation ↔ matrix

Once a basis is fixed the two languages are interchangeable, and every statement about a map has a matrix statement beside it. The point of reading them in parallel is to notice that neither side is primary — the matrix is not an approximation to the transformation, it is the transformation written in coordinates.

7entries

The correspondence itself3

1

The map and its matrix§ 1

unique

m \times n

matrix per linear

T

T(\mathbf{x}) = A\mathbf{x}

Every linear map from

\mathbb{R}^n

to

\mathbb{R}^m

is matrix multiplication and nothing else — there is no more general kind of linear map hiding elsewhere. The uniqueness matters as much as the existence: two different matrices cannot represent the same map in the same basis.

2

Columns are images of basis vectors§ 2

how the matrix is built

T(\mathbf{e}_j) = j

-th column of

A

The whole construction in one line. Feed the map each basis vector in turn and record what comes back — the results are the columns. This is why knowing a linear map on a basis is knowing it everywhere.

3

Dimensions§ 2

domain

\mathbb{R}^n

, codomain

\mathbb{R}^m

n

columns,

m

rows

Columns count inputs, rows count outputs. A linear operator has

V = W

, so its matrix is square — which is the only case where eigenvalues, determinants and inverses are even defined.

Operations translate2

4

Composition§ 6

note the order

S \circ T \leftrightarrow BA

Applying

T

then

S

corresponds to the product

BA

, not

AB

— the matrix acting first stands on the right, because it meets the vector first. This is the reason matrix multiplication is defined the way it is rather than entry by entry.

5

Identity and inverse§ 7

T

bijective

\iff

A

invertible

I \leftrightarrow I_n

,

T^{-1} \leftrightarrow A^{-1}

The identity transformation is the identity matrix in every basis — the one matrix a change of basis cannot alter. Invertibility transfers exactly, which is what makes the invertibility equivalence a statement about transformations too.

Subspaces translate2

6

Image and kernel§ 8

always

\operatorname{im}(T) \leftrightarrow \operatorname{Col}(A)

,

\ker(T) \leftrightarrow \operatorname{Null}(A)

The reachable outputs are the column space; the collapsed inputs are the null space. So rank-nullity is a statement about transformations that happens to be provable by row reduction.

7

Isomorphism§ 4

\dim V = \dim W

, both finite

invertible square matrix

Two spaces of the same finite dimension are isomorphic, and the isomorphism is a change of coordinates. This is why

\mathcal{P}_3

and

\mathbb{R}^{2\times 2}

are the same object twice — both four-dimensional, so both are

\mathbb{R}^4

in disguise.

The dictionary is exact and it is one-to-one, but only once a basis is chosen. Change the basis and the same transformation gets a different matrix — which is what makes similarity the right notion of sameness for matrices, and why properties surviving a basis change are the ones worth naming.

Transformation ↔ matrix·/linear-algebra/transformations/matrix-representationLearn Math Class

The composition row is where the dictionary explains something rather than merely recording it. Applying

T

and then

S

corresponds to the product

BA

— the matrix acting first written on the right, because it is the one the vector meets first. Matrix multiplication is defined as it is precisely so that this correspondence holds; the row-by-column rule is a consequence of wanting composition to translate, not an arbitrary convention.

The dictionary is exact, and it is also basis-dependent. Fix a basis and the correspondence is one-to-one; change the basis and the same transformation acquires a different matrix. That is the whole reason similarity exists as a notion — two matrices related by

P^{-1}AP

are the same map seen from two places, which is why the properties worth naming are the ones that survive the change.

Matrix Representation FAQ

Is $[T]_{\mathcal{C} \leftarrow \mathcal{B}}$ the same thing as $T$ ?

+

No. The map

T

is basis-free, while the bracketed object is one of infinitely many arrays that represent it, each attached to a particular choice of basis at both ends. Texts that write

T

for both are leaning on a fixed basis the reader is expected to remember, and the gap reappears the moment the basis changes.Read more →

Why do the images of the basis vectors go in columns rather than rows?

+

Because the matrix multiplies a column vector sitting on its right, so the arithmetic only works out if the images are stacked down columns. Reading them across rows instead produces the transpose and breaks

T(\mathbf{x}) = A\mathbf{x}

entirely. The convention is fixed by the multiplication rule, not by preference.Read more →

When can you drop the basis subscripts and just write $[T]$ ?

+

Only when both spaces carry the standard basis, which makes it the standard matrix, the one case where a map has a single uncontested array. An operator mapping a space to itself uses one basis twice and is written

[T]_{\mathcal{B}}

rather than repeating the label on both sides of the arrow.Read more →

Matric Representation of Linear Transformation

Encoding a Transformation as a Matrix

Every Linear Map from Rⁿ to Rᵐ Is Matrix Multiplication

Matrix Representation Notation

Matrix Representation Notation

Constructing the Standard Matrix

Worked Example

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

Transformation ↔ matrix

Matrix Representation FAQ

Is $[T]_{\mathcal{C} \leftarrow \mathcal{B}}$ the same thing as $T$ ?

Why do the images of the basis vectors go in columns rather than rows?

When can you drop the basis subscripts and just write $[T]$ ?

Matric Representation of Linear Transformation

Encoding a Transformation as a Matrix

Every Linear Map from Rⁿ to Rᵐ Is Matrix Multiplication

Matrix Representation Notation

Matrix Representation Notation

Constructing the Standard Matrix

Worked Example

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

Transformation ↔ matrix

Matrix Representation FAQ

Is [T]C←B[T]_{\mathcal{C} \leftarrow \mathcal{B}}[T]C←B​ the same thing as TTT?

Why do the images of the basis vectors go in columns rather than rows?

When can you drop the basis subscripts and just write [T][T][T]?

Is $[T]_{\mathcal{C} \leftarrow \mathcal{B}}$ the same thing as $T$ ?

When can you drop the basis subscripts and just write $[T]$ ?