Vector Spaces

The eight axioms

A vector space V over a field F (typically the reals ℝ or complex numbers ℂ) consists of:

• A set V (whose elements are called vectors).
• An operation + (addition) on V.
• An operation · (scalar multiplication) where scalars come from F.

satisfying eight axioms:

Axiom	Statement
Closure (addition)	u + v is in V
Commutativity	u + v = v + u
Associativity (addition)	(u + v) + w = u + (v + w)
Identity	There exists 0 such that v + 0 = v
Inverses	For each v, there exists −v such that v + (−v) = 0
Closure (scalar mult)	c · v is in V
Distributivity (over scalars)	c · (u + v) = c·u + c·v
Distributivity (over vectors) and Associativity	(c + d)·v = c·v + d·v; c·(d·v) = (cd)·v; 1·v = v

If a set with two operations satisfies all eight, it's a vector space. ℝⁿ does. So do many other things.

Examples of vector spaces

1. ℝⁿ — n-dimensional real space

The canonical example. Vectors are n-tuples of reals; addition and scalar multiplication are componentwise. ℝ¹ is the real line; ℝ² is the plane; ℝ³ is 3D space.

2. The polynomial space P_n

The set of polynomials of degree ≤ n with real coefficients. (3x² + 2x + 1) + (x² − 5) = 4x² + 2x − 4 — addition is well-defined. 2 · (3x² + 1) = 6x² + 2 — scalar multiplication too. Dimension n + 1; basis is {1, x, x², ..., x^n}.

3. The space of continuous functions C[0, 1]

All continuous real-valued functions on [0, 1]. Addition (f + g)(x) = f(x) + g(x). Scalar multiplication (c·f)(x) = c · f(x). The result is again continuous, satisfying closure. This is an infinite-dimensional vector space — no finite basis works.

4. The matrix space M_{m×n}

All m×n matrices with real entries. Addition is entry-wise; scalar multiplication scales all entries. Dimension m·n.

5. The complex numbers ℂ

The complex numbers form a vector space over the reals. Dimension 2 (basis {1, i}). The complex numbers can also be viewed as a vector space over themselves, in which case dimension is 1.

6. Polynomial sequences

Sequences (a₀, a₁, a₂, ...) of real numbers, with componentwise operations. Infinite dimensional. Subspace — those that are eventually 0 (finite-degree polynomial coefficient sequences).

Subspaces — vector spaces inside vector spaces

A subspace W of V is a subset that's closed under V's operations:

• 0 ∈ W (W is non-empty and contains the zero vector).
• u + v ∈ W whenever u, v ∈ W (closed under addition).
• c·v ∈ W whenever v ∈ W and c is a scalar (closed under scalar multiplication).

Examples:

• {0} is the trivial subspace — the smallest possible.
• Lines through the origin in ℝ² are subspaces.
• Planes through the origin in ℝ³ are subspaces.
• The set of polynomials with f(0) = 0 is a subspace of P_n.
• The kernel and image of any linear transformation are subspaces.

Subsets NOT through the origin (e.g., a line not passing through 0) are not subspaces — they fail the "contains 0" condition.

Span and linear independence

The span of a set of vectors {v₁, ..., vₖ} is the set of all linear combinations c₁v₁ + ... + cₖvₖ. The span is always a subspace.

A set is linearly independent if the only way to write 0 as c₁v₁ + ... + cₖvₖ is with all cᵢ = 0. Equivalently — no vector in the set is a linear combination of the others.

A basis is a linearly independent set whose span is the entire space. Every vector space has a basis (assuming the axiom of choice for infinite-dimensional spaces). Every basis has the same number of elements — that number is the dimension.

Dimension

Vector space	Dimension	Standard basis
ℝⁿ	n	{e₁, e₂, ..., eₙ} (standard basis vectors)
ℂⁿ over ℂ	n	{e₁, e₂, ..., eₙ}
ℂⁿ over ℝ	2n	{e₁, ie₁, ..., eₙ, ieₙ}
P_n (polynomials of degree ≤ n)	n + 1	{1, x, x², ..., x^n}
M_{m × n} (m × n matrices)	m·n	Matrices with one 1 entry, rest 0
C[a, b] (continuous functions on [a, b])	infinite	No finite basis
ℓ² (square-summable sequences)	infinite — Hilbert space	(1, 0, 0, ...), (0, 1, 0, ...), ...

Finite-dimensional vector spaces over the same field are isomorphic if they have the same dimension. So ℝⁿ "is" any other n-dimensional real vector space (after choosing a basis). The abstraction lets you transfer linear-algebra results among them.

Why the abstraction matters

• Theorems prove once, apply everywhere. The rank-nullity theorem, eigenvalue theorems, change-of-basis formulas — proven for general vector spaces, immediately applicable to functions, polynomials, matrices, etc.
• Function spaces enable PDE and quantum theory. Quantum states, signal processing, differential equations all live in function spaces. The vector-space framework provides the language.
• Linear-algebra reasoning extends to "vectors" that don't look like arrows. Polynomials, matrices, and operators all behave linearly; vector-space axioms encode this uniformly.
• Abstract definitions reveal essential structure. The 8 axioms strip away "what does this thing look like?" to focus on "how does it combine?" The latter is what matters for linear methods.

JavaScript — checking vector-space axioms

// Verify ℝⁿ is a vector space (sample checks)
function add(u, v) { return u.map((x, i) => x + v[i]); }
function scale(c, v) { return v.map(x => c * x); }
function zero(n) { return Array(n).fill(0); }

// Test commutativity
const a = [1, 2, 3], b = [4, 5, 6];
console.log(JSON.stringify(add(a, b)) === JSON.stringify(add(b, a)));  // true

// Test distributivity
const c = 5;
const lhs = scale(c, add(a, b));
const rhs = add(scale(c, a), scale(c, b));
console.log(JSON.stringify(lhs) === JSON.stringify(rhs));  // true

// Test linear independence — does 2·u + 3·v = 0 force u = v = 0?
function isLinearlyIndependent(vectors) {
  // For small examples: try all small integer combinations
  // For real applications: compute matrix rank
  // ...
}

Where vector spaces appear

• Linear algebra textbooks. The setting for linear transformations, matrix theory, eigenvalues, and SVD.
• Functional analysis. Hilbert spaces (infinite-dim with inner product) and Banach spaces (with a norm) — the function-space generalizations.
• Quantum mechanics. Wavefunctions live in a Hilbert space. Operators are linear transformations on it.
• Differential equations. Solutions to linear ODEs form a vector space; the dimension equals the order of the ODE. Boundary-value problems are also linear.
• Machine learning. Word embeddings, image features, neural network activations all live in vector spaces. Linear methods (SVM, PCA) operate on them.
• Computer graphics. 3D space ℝ³, 4D homogeneous coordinates, color spaces (RGB is ℝ³, but with constraints).

Common mistakes

• Confusing vector space with ℝⁿ. ℝⁿ is one example. Polynomials, functions, matrices are also vector spaces. The abstraction is broader than the concrete.
• Confusing affine subset with subspace. A line through (1, 0) parallel to the x-axis is NOT a subspace — it doesn't contain 0. Subspaces always pass through the origin.
• Forgetting that all 8 axioms are required. Skipping verification of associativity or commutativity has bitten people. Always check all eight when proving something is a vector space.
• Mixing up basis vs spanning set vs linearly independent set. Basis = both linearly independent AND spanning. A spanning set isn't always a basis (might be too big). A linearly independent set isn't always a basis (might not span). Both conditions together are required.
• Trying to define dimension in non-vector-space settings. Sets that aren't vector spaces don't have linear-algebra dimension. Don't try to apply rank, span, or basis to non-linear objects.
• Forgetting field choices. ℂⁿ is 2n-dimensional over ℝ but n-dimensional over ℂ. The base field matters when discussing dimension.