Vector Spaces

Abstract sets closed under addition and scalar multiplication — the structural framework that unifies vectors, functions, and matrices.

Vector addition: a + b = a + b

Definition

A vector space (or linear space) is a set $V$ of objects (called vectors) equipped with two operations — addition and scalar multiplication — satisfying certain axioms.

Key axioms: for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and scalars $a, b$ :

Closure: $\mathbf{u} + \mathbf{v} \in V$ and $a\mathbf{u} \in V$
Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
Zero vector: $\exists\, \mathbf{0}: \mathbf{u} + \mathbf{0} = \mathbf{u}$
Distributivity: $a(\mathbf{u}+\mathbf{v}) = a\mathbf{u} + a\mathbf{v}$

Example vector spaces: $\mathbb{R}^n$ , polynomials of degree $\leq n$ , functions on $[0,1]$ , matrices of size $m \times n$ .

Key properties

Closed under addition and scalar multiplication: combining vectors never leaves the space
A unique zero vector $\mathbf{0}$ exists, satisfying $\mathbf{u}+\mathbf{0}=\mathbf{u}$ for every $\mathbf{u}$
Every vector $\mathbf{u}$ has an additive inverse $-\mathbf{u}$ with $\mathbf{u}+(-\mathbf{u})=\mathbf{0}$
Scalar multiplication distributes over both vector addition and scalar addition

Common mistakes

Forgetting to check closure: a subset that "looks like" a vector space can still fail if it's not closed under addition or scaling (e.g. vectors with all-positive entries fail closure under scalar multiplication by $-1$ )
Confusing a vector space with $\mathbb{R}^n$ specifically: polynomials, matrices, and functions are all valid "vectors" too — the axioms, not the shape of the object, define a vector space

Polynomials as vectors

The set of polynomials $p(x) = a_0 + a_1 x + a_2 x^2$ of degree $\leq 2$ is a vector space. You can add polynomials and scale them. The "zero vector" is the zero polynomial $0 + 0x + 0x^2$ . The "dimension" is 3 — you need 3 coefficients to specify any element.

Try it

Is the set of all $n \times n$ matrices with trace = 0 a vector space? Check closure under addition and scalar multiplication.

Solution

Yes. If $\text{tr}(A) = 0$ and $\text{tr}(B) = 0$ , then $\text{tr}(A+B) = \text{tr}(A)+\text{tr}(B) = 0$ — closed under addition. If $\text{tr}(A) = 0$ , then $\text{tr}(cA) = c\,\text{tr}(A) = 0$ — closed under scalar multiplication. The zero matrix has trace 0. All axioms hold.

This is a subspace of the vector space of all $n\times n$ matrices. It has dimension $n^2 - 1$ .

Related concepts

Algebra· Linear Algebra

Linear IndependenceWhen no vector in a set can be written as a combination of the others — the condition that makes a set of vectors non-redundant.

Algebra· Linear Algebra

Span and BasisThe set of all vectors reachable by combining a given collection, and when that collection is minimal — the concept of a basis.

Algebra· Linear Algebra

Linear TransformationsFunctions between vector spaces that preserve addition and scaling — and why every such function is represented by a matrix.