Vector Space

Definition

A vector space (or linear space) over a field F is a set V together with two operations:

Vector Addition: A binary operation that combines two vectors to produce another vector
Scalar Multiplication: An operation that combines a scalar from F with a vector to produce another vector

These operations must satisfy eight axioms for all vectors u, v, w ∈ V and all scalars a, b ∈ F.

The Eight Axioms

Associativity of Addition

u + (v + w) = (u + v) + w

Commutativity of Addition

u + v = v + u

Identity Element of Addition

There exists a zero vector 0 ∈ V such that:

v + 0 = v

Inverse Elements of Addition

For every v ∈ V, there exists -v ∈ V such that:

v + (-v) = 0

Compatibility of Scalar Multiplication

a(bv) = (ab)v

Identity Element of Scalar Multiplication

1v = v

Where 1 is the multiplicative identity in F.

Distributivity of Scalar Multiplication over Vector Addition

a(u + v) = au + av

Distributivity of Scalar Multiplication over Field Addition

(a + b)v = av + bv

Examples of Vector Spaces

Euclidean Space Rⁿ

The set of all n-tuples of real numbers:

Rⁿ = {(x₁, x₂, ..., xₙ) | xᵢ ∈ R}

Operations are component-wise addition and scalar multiplication.

Polynomial Space P(x)

The set of all polynomials with real coefficients:

P(x) = {a₀ + a₁x + a₂x² + ... + aₙxⁿ | aᵢ ∈ R, n ≥ 0}

Matrix Space Mₘₓₙ(R)

The set of all m × n matrices with real entries.

Function Space C[a,b]

The set of all continuous real-valued functions on interval [a,b].

Important Concepts

Linear Combination

A vector w is a linear combination of vectors v₁, v₂, ..., vₙ if:

w = a₁v₁ + a₂v₂ + ... + aₙvₙ

where a₁, a₂, ..., aₙ are scalars.

Linear Independence

Vectors v₁, v₂, ..., vₙ are linearly independent if:

a₁v₁ + a₂v₂ + ... + aₙvₙ = 0 ⇒ a₁ = a₂ = ... = aₙ = 0

Basis

A basis is a linearly independent set of vectors that spans the entire vector space. Every vector in the space can be written uniquely as a linear combination of basis vectors.

Dimension

The dimension of a vector space is the number of vectors in any basis for the space.

Subspaces

A subspace W of a vector space V is a subset that is itself a vector space under the same operations. To verify W is a subspace:

0 ∈ W (contains zero vector)
Closed under addition: u, v ∈ W ⇒ u + v ∈ W
Closed under scalar multiplication: v ∈ W, a ∈ F ⇒ av ∈ W

Examples

Example 1: Verify that R² is a vector space.

Solution: Let u = (u₁, u₂), v = (v₁, v₂), w = (w₁, w₂) ∈ R² and a, b ∈ R.

Addition: u + v = (u₁ + v₁, u₂ + v₂) ∈ R² ✓
Commutativity: u + v = (u₁ + v₁, u₂ + v₂) = (v₁ + u₁, v₂ + u₂) = v + u ✓
Zero vector: 0 = (0, 0), so u + 0 = (u₁, u₂) = u ✓
Scalar multiplication: au = (au₁, au₂) ∈ R² ✓
All axioms are satisfied.

Example 2: Determine if W = {(x, y) ∈ R² | x + y = 0} is a subspace of R².

Solution:
1. Zero vector: (0, 0) satisfies 0 + 0 = 0, so 0 ∈ W ✓
2. Closure under addition: If (x₁, y₁), (x₂, y₂) ∈ W, then x₁ + y₁ = 0 and x₂ + y₂ = 0.
Their sum: (x₁ + x₂, y₁ + y₂). Check: (x₁ + x₂) + (y₁ + y₂) = (x₁ + y₁) + (x₂ + y₂) = 0 + 0 = 0 ✓
3. Closure under scalar multiplication: If (x, y) ∈ W, then x + y = 0.
For any scalar c: c(x, y) = (cx, cy). Check: cx + cy = c(x + y) = c(0) = 0 ✓
Therefore, W is a subspace of R².

Example 3: Show that vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for R³.

Solution:
1. Linear independence: Suppose a(1,0,0) + b(0,1,0) + c(0,0,1) = (0,0,0)
This gives (a, b, c) = (0, 0, 0), so a = b = c = 0. The vectors are linearly independent. ✓
2. Span: Any vector (x, y, z) ∈ R³ can be written as x(1,0,0) + y(0,1,0) + z(0,0,1). ✓
Therefore, these vectors form the standard basis for R³, and dim(R³) = 3.

Applications

Physics: Representing forces, velocities, and displacements in mechanics
Computer Graphics: 3D transformations, rotations, and projections
Engineering: Signal processing, control theory, and structural analysis
Data Science: Principal Component Analysis (PCA) and dimensionality reduction
Quantum Mechanics: State vectors in Hilbert spaces
Economics: Input-output models and optimization problems