Vector Spaces

The abstract setting where addition and scalar multiplication obey a short list of axioms, unifying vectors, functions, and more.

The abstraction

A vector space over a field of scalars (usually the real or complex numbers) is a set equipped with two operations, vector addition and scalar multiplication, that satisfy eight axioms. These require addition to be commutative and associative, guarantee a zero vector and additive inverses, and make scalar multiplication distribute over both kinds of addition.

The eight axioms in brief

• addition is commutative and associative
• there is a zero vector, and every vector has a negative
• scalar multiplication is associative: a(bv) = (ab)v
• the scalar 1 acts trivially: 1v = v
• two distributive laws link the operations

Examples beyond arrows

The point of the abstraction is that many different objects satisfy these rules. R^n is the familiar example, but so are the set of all polynomials of degree at most n, the set of continuous functions on an interval, the set of m-by-n matrices, and the solution set of a homogeneous linear differential equation. Every theorem proved from the axioms applies to all of them at once.

Subspaces

A subspace is a subset that is itself a vector space under the inherited operations. A nonempty subset is a subspace exactly when it is closed under addition and scalar multiplication. Lines and planes through the origin are subspaces of R^3; the column space and null space of a matrix are subspaces.

Function spaces are where this abstraction earns its keep in physics. Fields, wavefunctions, and signals all live in infinite-dimensional vector spaces, and the same concepts of basis, projection, and orthogonality carry over with inner products replacing dot products.

python

# Polynomials of degree <= 2 form a 3-dim vector space
# with basis {1, x, x^2}. A polynomial's coordinates:
p = [1.0, -2.0, 0.5]   # 1 - 2x + 0.5 x^2
print('coords in {1, x, x^2}:', p)

Discretizing a field on a grid replaces an infinite-dimensional function space with a large finite-dimensional one, the step that makes numerical simulation of continuous physics possible.