Linear Maps: Structure-Preserving Maps Between Vector Spaces

The rank–nullity theorem states that dim(ker f) + dim(im f) = dim(V) for any linear map f : V → W. We prove this central result after developing kernels, images, and injectivity/surjectivity criteria, then use it to classify finite-dimensional vector spaces up to isomorphism.

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March 1, 2026

1. Definition and Examples of Linear Maps

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 followers·107 articles

Linear Maps: Structure-Preserving Maps Between Vector Spaces

Folio Official

March 1, 2026

1. Definition and Examples of Linear Maps

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 followers·107 articles

Linear Maps: Structure-Preserving Maps Between Vector Spaces

1. Definition and Examples of Linear Maps

2. Kernel and Image

3. The Rank–Nullity Theorem

4. The Space of Linear Maps

Share your expertise with the world

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Linear Maps: Structure-Preserving Maps Between Vector Spaces

1. Definition and Examples of Linear Maps

2. Kernel and Image

3. The Rank–Nullity Theorem

4. The Space of Linear Maps

Share your expertise with the world

More from Folio Official

Matrices and Representation of Linear Maps

The Determinant: A Scalar Invariant of Square Matrices

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