Vector Spaces

Overview

A vector space is a set $V$ with two operations (addition and scalar multiplication) that satisfy specific axioms. Vector spaces generalize the properties of $\mathbb{R}^n$.

Definition

A vector space over a field $F$ (usually $\mathbb{R}$) is a set $V$ with:

• Addition: $V \times V \to V$
• Scalar multiplication: $F \times V \to V$

Vector Space Axioms

Axiom	Property
A1	$\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ (commutativity)
A2	$(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$ (associativity)
A3	$\exists \mathbf{0} \in V: \mathbf{v} + \mathbf{0} = \mathbf{v}$ (identity)
A4	$\exists -\mathbf{v} \in V: \mathbf{v} + (-\mathbf{v}) = \mathbf{0}$ (inverse)
S1	$c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$
S2	$(c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v}$
S3	$c(d\mathbf{v}) = (cd)\mathbf{v}$
S4	$1\mathbf{v} = \mathbf{v}$

Common Examples

$\mathbb{R}^n$ - n-dimensional Real Space

The set of all n-tuples of real numbers:

$\mathbb{R}^n = \{(x_1, x_2, \ldots, x_n) : x_i \in \mathbb{R}\}$

Polynomials $P_n$

The set of polynomials of degree $\leq n$:

$P_n = \{a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n : a_i \in \mathbb{R}\}$

Matrices $M_{m \times n}$

The set of $m \times n$ matrices with real entries.

Functions

The set of continuous functions on an interval.

Closure Properties

A vector space must be closed under:

Addition Closure

If $\mathbf{u}, \mathbf{v} \in V$, then $\mathbf{u} + \mathbf{v} \in V$

Scalar Multiplication Closure

If $\mathbf{v} \in V$ and $c \in F$, then $c\mathbf{v} \in V$

Verifying a Vector Space

To verify $V$ is a vector space:

1. Check closure under addition
2. Check closure under scalar multiplication
3. Verify all 8 axioms hold

Example: Verification

Verify $\mathbb{R}^2$ is a vector space

Let $\mathbf{u} = (u_1, u_2)$, $\mathbf{v} = (v_1, v_2) \in \mathbb{R}^2$, $c, d \in \mathbb{R}$

1. Closure (addition): $\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2) \in \mathbb{R}^2$ ✓
2. Closure (scalar): $c\mathbf{u} = (cu_1, cu_2) \in \mathbb{R}^2$ ✓
3. Identity: $\mathbf{0} = (0, 0)$ satisfies $\mathbf{u} + \mathbf{0} = \mathbf{u}$ ✓
4. Inverse: $-\mathbf{u} = (-u_1, -u_2)$ satisfies $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$ ✓
5. All axioms hold by properties of real numbers ✓

Non-Examples

The following are NOT vector spaces:

• Positive real numbers under usual addition (no additive inverse)
• Polynomials of exactly degree $n$ (not closed under addition)
• The set $\{0, 1\}$ under usual addition ($1 + 1 = 2 \notin \{0, 1\}$)

Properties of Vector Spaces

For any vector space $V$:

• $0\mathbf{v} = \mathbf{0}$ for any $\mathbf{v} \in V$
• $c\mathbf{0} = \mathbf{0}$ for any scalar $c$
• $(-1)\mathbf{v} = -\mathbf{v}$
• If $c\mathbf{v} = \mathbf{0}$, then $c = 0$ or $\mathbf{v} = \mathbf{0}$