Linear Maps: Structure-Preserving Maps Between Vector Spaces

We define linear maps and study their fundamental properties, including the kernel and image. The centerpiece is the rank--nullity theorem, which relates the dimensions of these subspaces. We also discuss the vector space of linear maps and the notion of isomorphism.

Folio Official

March 1, 2026

1 Definition and Basic Examples

Definition 1 (Linear map).

Let

V

and

W

be vector spaces over

K

. A map

T : V \to W

is called a linear map (or linear transformation) if it satisfies the following two conditions:

$T (u + v) = T (u) + T (v)$ (preservation of addition).
$T (a v) = a T (v)$ (preservation of scalar multiplication).

Remark 2.

The two conditions can be consolidated into one:

T (a u + b v) = a T (u) + b T (v)

for all

a, b \in K

and

u, v \in V

Example 3.

The rotation matrix $R_{θ} : R^{2} \to R^{2}$ defines a linear map.
The differentiation operator $D : K [x]_{\leq n} \to K [x]_{\leq n - 1}$ , defined by $D f = f^{'}$ , is linear.
The definite integral $I : C [a, b] \to R$ , given by $I (f) = \int_{a}^{b} f (x) d x$ , is linear.
The map $T : R^{2} \to R^{2}$ defined by $T (v) = v + (1, 0)$ is not linear, since $T (0) \neq = 0$ .

Theorem 4.

Every linear map

T : V \to W

sends the zero vector to the zero vector:

T (0_{V}) = 0_{W}

Proof.

T (0) = T (0 \cdot 0) = 0 \cdot T (0) = 0

. □

Theorem 5 (A linear map is determined by its action on a basis).

Let

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

be a basis for

V

. For any choice of vectors

w_{1}, \dots, w_{n} \in W

, there exists a unique linear map

T : V \to W

satisfying

T (v_{i}) = w_{i}

for

i = 1, \dots, n

Proof.

For

v = \sum a_{i} v_{i}

, define

T (v) = \sum a_{i} w_{i}

. The unique representation of each

v

with respect to the basis ensures that

T

is well-defined, and linearity is verified by direct computation. Uniqueness holds because the images of the basis vectors completely determine

T

. □

2 Kernel and Image

Definition 6 (Kernel).

The kernel (or null space) of a linear map

T : V \to W

is the set

ker T = {v \in V ∣ T (v) = 0} .

Definition 7 (Image).

The image (or range) of a linear map

T : V \to W

is the set

Im T = {T (v) ∣ v \in V} = {w \in W ∣ \exists v \in V, T (v) = w} .

Theorem 8.

ker T

is a subspace of

V

, and

Im T

is a subspace of

W

Proof.

For the kernel: if

u, v \in ker T

, then

T (u + v) = T (u) + T (v) = 0

, so

u + v \in ker T

. Also,

T (a v) = a T (v) = a 0 = 0

, so

a v \in ker T

For the image: if

T (u), T (v) \in Im T

, then

T (u) + T (v) = T (u + v) \in Im T

. Also,

a T (v) = T (a v) \in Im T

. □

Theorem 9 (Injectivity and the kernel).

A linear map

T

is injective if and only if

ker T = {0}

Proof.

(\Rightarrow)

v \in ker T

, then

T (v) = 0 = T (0)

, and injectivity gives

v = 0

(\Leftarrow)

T (u) = T (v)

, then

T (u - v) = 0

, so

u - v \in ker T = {0}

, whence

u = v

. □

3 The Rank–Nullity Theorem

Definition 10.

The dimension

dim ker T

is called the nullity of

T

, and

dim Im T

is called the rank of

T

Theorem 11 (Rank–nullity theorem).

Let

V

be a finite-dimensional vector space and let

T : V \to W

be a linear map. Then

dim V = dim ker T + dim Im T .

Proof.

Let

dim ker T = r

and choose a basis

{u_{1}, \dots, u_{r}}

for

ker T

. Extend this to a basis

{u_{1}, \dots, u_{r}, v_{1}, \dots, v_{s}}

for

V

, where

r + s = dim V

We claim that

{T (v_{1}), \dots, T (v_{s})}

is a basis for

Im T

Spanning. Let

w \in Im T

. Then

w = T (v)

for some

v = \sum a_{i} u_{i} + \sum b_{j} v_{j}

. Since each

u_{i} \in ker T

, we have

T (v) = \sum b_{j} T (v_{j})

Linear independence. Suppose

\sum b_{j} T (v_{j}) = 0

. Then

T (\sum b_{j} v_{j}) = 0

, so

\sum b_{j} v_{j} \in ker T

. This means

\sum b_{j} v_{j} = \sum a_{i} u_{i}

for some scalars

a_{i}

, i.e.

\sum a_{i} u_{i} - \sum b_{j} v_{j} = 0

. Since

{u_{1}, \dots, u_{r}, v_{1}, \dots, v_{s}}

is a basis, all coefficients are zero. In particular,

b_{j} = 0

for each

j

Therefore

dim Im T = s = dim V - dim ker T

. □

Example 12.

Let

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

be defined by

T (x, y, z) = (x + y, y + z)

. One checks that

ker T = {(t, - t, t) ∣ t \in R}

, so

dim ker T = 1

. By the rank–nullity theorem,

dim Im T = 3 - 1 = 2 = dim R^{2}

. Hence

T

is surjective.

4 The Space of Linear Maps

Definition 13.

Let

V

and

W

be vector spaces over

K

. The set of all linear maps from

V

W

is denoted

Hom_{K} (V, W)

. Equipped with pointwise addition

(T + S) (v) = T (v) + S (v)

and scalar multiplication

(a T) (v) = a T (v)

, it is itself a vector space over

K

Theorem 14.

dim V = n

and

dim W = m

, then

dim Hom_{K} (V, W) = mn

Proof.

By the theorem on determination by basis images, every

T \in Hom (V, W)

is determined by the images of a basis of

V

— that is, by

n

vectors in

W

, each having

m

coordinates. This correspondence is an isomorphism

Hom_{K} (V, W) ≅ M_{m \times n} (K)

, and

dim M_{m \times n} (K) = mn

. □

Definition 15 (Isomorphism).

A linear map

T : V \to W

that is bijective is called an isomorphism. When an isomorphism exists, we write

V ≅ W

Theorem 16.

For finite-dimensional vector spaces

V

and

W

V ≅ W

if and only if

dim V = dim W

Linear Algebra Algebra Textbook Linear Maps Kernel and Image Rank--Nullity Theorem

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 フォロワー·105 記事

Linear Maps: Structure-Preserving Maps Between Vector Spaces

1 Definition and Basic Examples

2 Kernel and Image

3 The Rank–Nullity Theorem

4 The Space of Linear Maps

あなたの専門知識を世界に発信しよう

Folio Officialの他の記事

Monetizing Your Articles — How to Earn Revenue Safely with Stripe Connect

Mastering Theorem Environments — theorem, definition, proof, and Friends

Combinatorial Designs and Latin Squares: Balanced Arrangements

Burnside's Lemma and P\'olya Enumeration: Counting Under Symmetry