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The Definition of a Vector Space

Let $V$ be a nonempty set. If addition and scalar multiplication are defined on $V$ and satisfy the following 8 axioms, then $V$ is called a vector space (or a linear space).

The addition axioms:

Associativity: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
Commutativity: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
The zero element: there exists $\vec{0} \in V$ such that $\vec{a} + \vec{0} = \vec{a}$
The inverse element: for every $\vec{a} \in V$ there exists $-\vec{a} \in V$ such that $\vec{a} + (-\vec{a}) = \vec{0}$

The scalar multiplication axioms: 5. Distributivity: $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$ 6. Distributivity: $(k + l)\vec{a} = k\vec{a} + l\vec{a}$ 7. Associativity: $(kl)\vec{a} = k(l\vec{a})$ 8. The identity element: $1 \cdot \vec{a} = \vec{a}$

A Geometric Interpretation of a Vector Space

A vector space can be viewed as a set of “vectors” on which addition and scalar multiplication can be carried out, just as you can operate on vectors in the plane or in space.

Common Vector Spaces

Vector Spaces over the Real Numbers

$\mathbb{R}^n = \{ (a_1, a_2, \dots, a_n) \mid a_i \in \mathbb{R} \}$

This is the most common vector space, and it includes:

$\mathbb{R}^1$ : the number line
$\mathbb{R}^2$ : the plane
$\mathbb{R}^3$ : three-dimensional space

The Space of Functions

$P_n = \{ a_0 + a_1x + \dots + a_nx^n \mid a_i \in \mathbb{R} \}$

The set of all polynomials of degree at most $n$ .

The Space of Matrices

$M_{m \times n} = \{ A = (a_{ij})_{m \times n} \mid a_{ij} \in \mathbb{R} \}$

The set of all $m \times n$ real matrices.

The Basic Properties of a Vector Space

Theorem1

In a vector space $V$ , the following properties hold:

The zero vector is unique: the zero vector is unique
The negative vector is unique: the negative of each vector is unique
Scalar multiplication by zero: $0 \cdot \vec{a} = \vec{0}$ , $k \cdot \vec{0} = \vec{0}$
A property of scalar multiplication: $(-1) \cdot \vec{a} = -\vec{a}$

Exercises

Exercise 1

Verify that $\mathbb{R}^2$ forms a vector space.

Reference Answer

vector spaceverifying the axiomsa real vector space

Approach:

Verify that $\mathbb{R}^2$ satisfies the 8 axioms of a vector space.

Steps:

Associativity of addition: $(a,b) + ((c,d) + (e,f)) = ((a,b) + (c,d)) + (e,f)$ The left-hand side: $(a+c+e, b+d+f)$ The right-hand side: $(a+c+e, b+d+f)$ It holds
Commutativity of addition: $(a,b) + (c,d) = (c,d) + (a,b)$ Both sides are $(a+c, b+d)$ It holds
The zero vector: $(0,0) + (a,b) = (a,b)$ It holds
The negative vector: $(a,b) + (-a,-b) = (0,0)$ It holds
Distributivity over addition: $k((a,b) + (c,d)) = k(a+c, b+d) = (k(a+c), k(b+d))$ $k(a,b) + k(c,d) = (ka, kb) + (kc, kd) = (ka+kc, kb+kd)$ It holds
Distributivity over scalar addition: $(k+l)(a,b) = (k+l)a, (k+l)b$ $k(a,b) + l(a,b) = (ka, kb) + (la, lb) = (ka+la, kb+lb)$ It holds
Associativity of scalar multiplication: $(kl)(a,b) = (kl a, kl b)$ $k(l(a,b)) = k(la, lb) = (kla, klb)$ It holds
The identity element: $1 \cdot (a,b) = (a,b)$ It holds

Answer: $\mathbb{R}^2$ forms a vector space

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$V$	Set	Vector space V	A vector space
$\vec{a}, \vec{b}, \vec{c}$	Vector	vectors a, b, c	Elements of the vector space
$\vec{0}$	Zero vector	zero vector	The zero element of the vector space
$k, l$	Scalar	scalars k, l	Real numbers

Vector Spaces

Bases and Dimension

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The Definition of a Vector Space