Vector Spaces: Definitions and First Properties

A vector space has exactly one zero vector and every element has a unique additive inverse — these are consequences of the eight axioms, not assumptions. We prove these uniqueness results, establish the subspace criterion for verifying subspaces efficiently, and characterize when a sum of subspaces is a direct sum.

Folio Official

March 1, 2026

1. The Vector Space Axioms

Definition 1 (Vector space).

A vector space over a field

K

is a set

V

equipped with two operations — addition

+ : V \times V \to V

and scalar multiplication

\cdot : K \times V \to V

— satisfying the following eight axioms:

$u + v = v + u$ (commutativity of addition).
$(u + v) + w = u + (v + w)$ (associativity of addition).
There exists an element $0 \in V$ such that $v + 0 = v$ for all $v \in V$ (existence of a zero vector).
For every $v \in V$ , there exists $- v \in V$ such that $v + (- v) = 0$ (existence of additive inverses).
$a (b v) = (ab) v$ (associativity of scalar multiplication).
$1 \cdot v = v$ (identity element of scalar multiplication).
$a (u + v) = a u + a v$ (distributivity over vector addition).
$(a + b) v = a v + b v$ (distributivity over scalar addition).

Here

a, b \in K

and

u, v, w \in V

Throughout this chapter, we work over the field $K = R$ unless stated otherwise.

2. Fundamental Examples

Example 2 (

K^{n}

R^{n} = {(a_{1}, \dots, a_{n}) ∣ a_{i} \in R}

is a vector space over

R

under componentwise addition and scalar multiplication. The zero vector is

0 = (0, \dots, 0)

Example 3 (Polynomial spaces).

K [x]_{\leq n} = {a_{0} + a_{1} x + \dots + a_{n} x^{n} ∣ a_{i} \in K}

is a vector space over

K

under polynomial addition and scalar multiplication. The zero vector is the zero polynomial

0

, and

dim K [x]_{\leq n} = n + 1

Example 4 (Matrix spaces).

The set

M_{m \times n} (K)

of all

m \times n

matrices over

K

is a vector space under matrix addition and scalar multiplication. The zero vector is the zero matrix

O

, and

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

Example 5 (Function spaces).

The set

C [a, b]

of all real-valued continuous functions on the interval

[a, b]

forms a vector space over

R

under pointwise addition

(f + g) (x) = f (x) + g (x)

and scalar multiplication

(c f) (x) = c \cdot f (x)

. This is an infinite-dimensional vector space.

3. Uniqueness of the Zero Vector and Additive Inverses

Theorem 6 (Uniqueness of the zero vector).

The zero vector of a vector space

V

is unique.

Proof.

Suppose

0

and

0^{'}

are both zero vectors. Since

0

is a zero vector,

0^{'} + 0 = 0^{'}

. Since

0^{'}

is a zero vector,

0^{'} + 0 = 0

. Therefore

0^{'} = 0

. □

Theorem 7 (Uniqueness of additive inverses).

For each

v \in V

, the additive inverse

- v

is unique.

Proof.

Suppose

w_{1}

and

w_{2}

are both additive inverses of

v

. Then

w_{1} = w_{1} + 0 = w_{1} + (v + w_{2}) = (w_{1} + v) + w_{2} = 0 + w_{2} = w_{2}

. □

4. Basic Properties of Scalar Multiplication

Theorem 8.

In any vector space

V

, the following hold:

$0 \cdot v = 0$ (the scalar $0$ annihilates every vector).
$a \cdot 0 = 0$ (every scalar annihilates the zero vector).
$(- 1) v = - v$ .
$a v = 0 ⟹ a = 0$ or $v = 0$ .

Proof.

(1) We have

0 \cdot v = (0 + 0) v = 0 v + 0 v

. Subtracting

0 v

from both sides yields

0 = 0 v

(2) Similarly,

a \cdot 0 = a (0 + 0) = a 0 + a 0

. Subtracting

a 0

from both sides gives

0 = a 0

(3) We compute

v + (- 1) v = 1 \cdot v + (- 1) v = (1 + (- 1)) v = 0 v = 0

. By uniqueness of the additive inverse,

(- 1) v = - v

(4) Suppose

a \neq = 0

. Multiplying both sides of

a v = 0

a^{- 1}

gives

v = a^{- 1} 0 = 0

. □

5. Subspaces

Definition 9 (Subspace).

A nonempty subset

W

of a vector space

V

is called a subspace of

V

W

is itself a vector space under the operations inherited from

V

Theorem 10 (Subspace criterion).

Let

V

be a vector space and

W \subseteq V

a nonempty subset. Then

W

is a subspace of

V

if and only if the following two conditions hold:

$u, v \in W ⟹ u + v \in W$ .
$a \in K, v \in W ⟹ a v \in W$ .

Proof.

Necessity is immediate. For sufficiency, since

W \neq = \emptyset

, pick any

v \in W

. Setting

a = 0

in condition (2) gives

0 = 0 \cdot v \in W

. Setting

a = - 1

gives

- v \in W

. The remaining axioms (associativity, commutativity, distributivity, etc.) are inherited from

V

. □

Remark 11.

Conditions (1) and (2) can be consolidated into a single condition:

a, b \in K

and

u, v \in W

imply

a u + b v \in W

Example 12.

R^{3}

, the set

W = {(x, y, z) ∣ x + y + z = 0}

is a subspace. Indeed, if

u = (u_{1}, u_{2}, u_{3})

and

v = (v_{1}, v_{2}, v_{3})

lie in

W

, then

(u_{1} + v_{1}) + (u_{2} + v_{2}) + (u_{3} + v_{3}) = 0

, so

u + v \in W

. Closure under scalar multiplication is verified similarly.

Example 13.

The set

{(x, y, z) ∣ x + y + z = 1}

is not a subspace, since the zero vector

0 = (0, 0, 0)

does not belong to it.

6. Sum Spaces and Direct Sums

Definition 14 (Sum of subspaces).

Let

W_{1}

and

W_{2}

be subspaces of

V

. Their sum is defined as

W_{1} + W_{2} = {w_{1} + w_{2} ∣ w_{1} \in W_{1}, w_{2} \in W_{2}} .

This is the smallest subspace of

V

containing

W_{1} \cup W_{2}

Definition 15 (Direct sum).

If every vector

v \in W_{1} + W_{2}

can be written uniquely as

v = w_{1} + w_{2}

with

w_{i} \in W_{i}

, then the sum is called a direct sum and is denoted

W_{1} \oplus W_{2}

Theorem 16.

The sum

W_{1} + W_{2}

is a direct sum if and only if

W_{1} \cap W_{2} = {0}

Proof.

(\Rightarrow)

Let

v \in W_{1} \cap W_{2}

. Then

v = v + 0 = 0 + v

provides two decompositions. By uniqueness,

v = 0

(\Leftarrow)

Suppose

v = w_{1} + w_{2} = w_{1}^{'} + w_{2}^{'}

. Then

w_{1} - w_{1}^{'} = w_{2}^{'} - w_{2} \in W_{1} \cap W_{2} = {0}

, whence

w_{1} = w_{1}^{'}

and

w_{2} = w_{2}^{'}

. □

Theorem 17 (Dimension formula for subspaces).

For finite-dimensional subspaces

W_{1}, W_{2}

V

dim (W_{1} + W_{2}) = dim W_{1} + dim W_{2} - dim (W_{1} \cap W_{2}) .

In particular, if

W_{1} + W_{2}

is a direct sum, then

dim (W_{1} \oplus W_{2}) = dim W_{1} + dim W_{2}

Linear Algebra Algebra Textbook Vector Spaces Subspaces Direct Sums

Folio Official

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

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Vector Spaces: Definitions and First Properties

Folio Official

March 1, 2026

1. The Vector Space Axioms

Definition 1 (Vector space).

A vector space over a field

K

is a set

V

equipped with two operations — addition

+ : V \times V \to V

and scalar multiplication

\cdot : K \times V \to V

— satisfying the following eight axioms:

$u + v = v + u$ (commutativity of addition).
$(u + v) + w = u + (v + w)$ (associativity of addition).
There exists an element $0 \in V$ such that $v + 0 = v$ for all $v \in V$ (existence of a zero vector).
For every $v \in V$ , there exists $- v \in V$ such that $v + (- v) = 0$ (existence of additive inverses).
$a (b v) = (ab) v$ (associativity of scalar multiplication).
$1 \cdot v = v$ (identity element of scalar multiplication).
$a (u + v) = a u + a v$ (distributivity over vector addition).
$(a + b) v = a v + b v$ (distributivity over scalar addition).

Here

a, b \in K

and

u, v, w \in V

Throughout this chapter, we work over the field $K = R$ unless stated otherwise.

2. Fundamental Examples

Example 2 (

K^{n}

R^{n} = {(a_{1}, \dots, a_{n}) ∣ a_{i} \in R}

is a vector space over

R

under componentwise addition and scalar multiplication. The zero vector is

0 = (0, \dots, 0)

Example 3 (Polynomial spaces).

K [x]_{\leq n} = {a_{0} + a_{1} x + \dots + a_{n} x^{n} ∣ a_{i} \in K}

is a vector space over

K

under polynomial addition and scalar multiplication. The zero vector is the zero polynomial

0

, and

dim K [x]_{\leq n} = n + 1

Example 4 (Matrix spaces).

The set

M_{m \times n} (K)

of all

m \times n

matrices over

K

is a vector space under matrix addition and scalar multiplication. The zero vector is the zero matrix

O

, and

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

Example 5 (Function spaces).

The set

C [a, b]

of all real-valued continuous functions on the interval

[a, b]

forms a vector space over

R

under pointwise addition

(f + g) (x) = f (x) + g (x)

and scalar multiplication

(c f) (x) = c \cdot f (x)

. This is an infinite-dimensional vector space.

3. Uniqueness of the Zero Vector and Additive Inverses

Theorem 6 (Uniqueness of the zero vector).

The zero vector of a vector space

V

is unique.

Proof.

Suppose

0

and

0^{'}

are both zero vectors. Since

0

is a zero vector,

0^{'} + 0 = 0^{'}

. Since

0^{'}

is a zero vector,

0^{'} + 0 = 0

. Therefore

0^{'} = 0

. □

Theorem 7 (Uniqueness of additive inverses).

For each

v \in V

, the additive inverse

- v

is unique.

Proof.

Suppose

w_{1}

and

w_{2}

are both additive inverses of

v

. Then

w_{1} = w_{1} + 0 = w_{1} + (v + w_{2}) = (w_{1} + v) + w_{2} = 0 + w_{2} = w_{2}

. □

4. Basic Properties of Scalar Multiplication

Theorem 8.

In any vector space

V

, the following hold:

$0 \cdot v = 0$ (the scalar $0$ annihilates every vector).
$a \cdot 0 = 0$ (every scalar annihilates the zero vector).
$(- 1) v = - v$ .
$a v = 0 ⟹ a = 0$ or $v = 0$ .

Proof.

(1) We have

0 \cdot v = (0 + 0) v = 0 v + 0 v

. Subtracting

0 v

from both sides yields

0 = 0 v

(2) Similarly,

a \cdot 0 = a (0 + 0) = a 0 + a 0

. Subtracting

a 0

from both sides gives

0 = a 0

(3) We compute

v + (- 1) v = 1 \cdot v + (- 1) v = (1 + (- 1)) v = 0 v = 0

. By uniqueness of the additive inverse,

(- 1) v = - v

(4) Suppose

a \neq = 0

. Multiplying both sides of

a v = 0

a^{- 1}

gives

v = a^{- 1} 0 = 0

. □

5. Subspaces

Definition 9 (Subspace).

A nonempty subset

W

of a vector space

V

is called a subspace of

V

W

is itself a vector space under the operations inherited from

V

Theorem 10 (Subspace criterion).

Let

V

be a vector space and

W \subseteq V

a nonempty subset. Then

W

is a subspace of

V

if and only if the following two conditions hold:

$u, v \in W ⟹ u + v \in W$ .
$a \in K, v \in W ⟹ a v \in W$ .

Proof.

Necessity is immediate. For sufficiency, since

W \neq = \emptyset

, pick any

v \in W

. Setting

a = 0

in condition (2) gives

0 = 0 \cdot v \in W

. Setting

a = - 1

gives

- v \in W

. The remaining axioms (associativity, commutativity, distributivity, etc.) are inherited from

V

. □

Remark 11.

Conditions (1) and (2) can be consolidated into a single condition:

a, b \in K

and

u, v \in W

imply

a u + b v \in W

Example 12.

R^{3}

, the set

W = {(x, y, z) ∣ x + y + z = 0}

is a subspace. Indeed, if

u = (u_{1}, u_{2}, u_{3})

and

v = (v_{1}, v_{2}, v_{3})

lie in

W

, then

(u_{1} + v_{1}) + (u_{2} + v_{2}) + (u_{3} + v_{3}) = 0

, so

u + v \in W

. Closure under scalar multiplication is verified similarly.

Example 13.

The set

{(x, y, z) ∣ x + y + z = 1}

is not a subspace, since the zero vector

0 = (0, 0, 0)

does not belong to it.

6. Sum Spaces and Direct Sums

Definition 14 (Sum of subspaces).

Let

W_{1}

and

W_{2}

be subspaces of

V

. Their sum is defined as

W_{1} + W_{2} = {w_{1} + w_{2} ∣ w_{1} \in W_{1}, w_{2} \in W_{2}} .

This is the smallest subspace of

V

containing

W_{1} \cup W_{2}

Definition 15 (Direct sum).

If every vector

v \in W_{1} + W_{2}

can be written uniquely as

v = w_{1} + w_{2}

with

w_{i} \in W_{i}

, then the sum is called a direct sum and is denoted

W_{1} \oplus W_{2}

Theorem 16.

The sum

W_{1} + W_{2}

is a direct sum if and only if

W_{1} \cap W_{2} = {0}

Proof.

(\Rightarrow)

Let

v \in W_{1} \cap W_{2}

. Then

v = v + 0 = 0 + v

provides two decompositions. By uniqueness,

v = 0

(\Leftarrow)

Suppose

v = w_{1} + w_{2} = w_{1}^{'} + w_{2}^{'}

. Then

w_{1} - w_{1}^{'} = w_{2}^{'} - w_{2} \in W_{1} \cap W_{2} = {0}

, whence

w_{1} = w_{1}^{'}

and

w_{2} = w_{2}^{'}

. □

Theorem 17 (Dimension formula for subspaces).

For finite-dimensional subspaces

W_{1}, W_{2}

V

dim (W_{1} + W_{2}) = dim W_{1} + dim W_{2} - dim (W_{1} \cap W_{2}) .

In particular, if

W_{1} + W_{2}

is a direct sum, then

dim (W_{1} \oplus W_{2}) = dim W_{1} + dim W_{2}

Linear Algebra Algebra Textbook Vector Spaces Subspaces Direct Sums

Folio Official

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

1 followers·107 articles

Vector Spaces: Definitions and First Properties

1. The Vector Space Axioms

2. Fundamental Examples

3. Uniqueness of the Zero Vector and Additive Inverses

4. Basic Properties of Scalar Multiplication

5. Subspaces

6. Sum Spaces and Direct Sums

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Vector Spaces: Definitions and First Properties

1. The Vector Space Axioms

2. Fundamental Examples

3. Uniqueness of the Zero Vector and Additive Inverses

4. Basic Properties of Scalar Multiplication

5. Subspaces

6. Sum Spaces and Direct Sums

Share your expertise with the world

More from Folio Official

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