11  Vector Spaces

Ask most people who’ve taken linear algebra what a vector is, and the answer is “an arrow” or “a list of numbers.” Both are correct — and both are instances of something more general, in exactly the sense What Is a Mathematical Structure? described: mathematicians care less about what an object is than about what structure it carries. This chapter is that idea made fully concrete, on the structure this whole site has been building toward.

11.1 The problem, before the definition

Displacements in the plane add: walk 3 meters east then 4 meters north, and the result is the same as one diagonal walk — vector addition. You can also scale a displacement: walk twice as far in the same direction. That’s the entire motivating problem — describe quantities that add and scale — and it’s worth seeing it satisfied by objects that don’t look anything like arrows before writing down the axioms:

  • \(\mathbb{R}^n\) — ordered tuples, added componentwise, scaled componentwise. The example everyone starts with.
  • Polynomials of degree \(\leq n\) — \((x^2 + 3x) + (2x^2 - x) = 3x^2 + 2x\) is addition; \(5(x^2 + 3x) = 5x^2 + 15x\) is scaling. No arrows in sight.
  • \(m \times n\) matrices — added entrywise, scaled entrywise.
  • Functions on some domain — \((f+g)(x) := f(x) + g(x)\), \((cf)(x) := c \cdot f(x)\). The set of all real-valued functions on \([0,1]\) adds and scales exactly as validly as \(\mathbb{R}^n\) does.

These four look nothing alike — tuples, polynomials, grids of numbers, functions — and yet, as the definition below will make precise, they satisfy exactly the same axioms. From the perspective of vector-space theory, they are the same kind of object.

11.2 The abstraction

A vector space over a field \(F\) — Rings and Fields makes precise what “field” means; think \(\mathbb{R}\) or \(\mathbb{C}\), which Number Systems built — is a set \(V\) with two operations, vector addition \(V \times V \to V\) and scalar multiplication \(F \times V \to V\), satisfying:

  1. \((V, +)\) is an abelian group — closure, associativity, an additive identity \(\vec{0}\), additive inverses, and commutativity (see Groups; a vector space’s addition is required to be commutative, unlike a general group’s operation).
  2. Scalar associativity: \(a(bv) = (ab)v\).
  3. Distributivity over vector addition: \(a(u+v) = au + av\).
  4. Distributivity over scalar addition: \((a+b)v = av + bv\).
  5. Scalar identity: \(1 \cdot v = v\).

11.3 Why these axioms

Each one is earning its keep:

  • The abelian-group axioms on \(+\) guarantee vectors combine and undo cleanly, and specifically commutatively — without commutativity, “walk east then north” and “walk north then east” could land you in different places, which would make the displacement picture this chapter opened with simply false.
  • Scalar associativity (\(a(bv) = (ab)v\)) is what makes “scale by 6” and “scale by 2, then by 3” interchangeable — without it, scaling wouldn’t be a coherent single operation, just a family of unrelated ones indexed by which scalar you happened to pick first.
  • The two distributivity laws are what make addition and scaling interact sensibly rather than being two unrelated operations that happen to share a set. Drop \(a(u+v) = au+av\) and doubling a sum of displacements could differ from summing the doubled displacements — the geometric picture of “scaling stretches the whole plane uniformly” would fail.
  • The scalar identity (\(1 \cdot v = v\)) rules out a degenerate “solution” where scalar multiplication collapses everything to zero regardless of the scalar — without it, “scale by 1” wouldn’t have to mean “leave unchanged,” which would make “scalar multiplication” a contradiction in terms.

11.4 What follows from the definition

A linear combination of vectors \(v_1, \dots, v_k\) is any \(c_1 v_1 + \cdots + c_k v_k\). The set of all linear combinations of a given collection of vectors is their span. A collection of vectors is linearly independent if no one of them is a linear combination of the others — equivalently, the only way to combine them to get \(\vec{0}\) is with every coefficient zero. A basis is a linearly independent set that spans all of \(V\); every vector in \(V\) is then a unique linear combination of basis vectors. That uniqueness is what lets you represent an abstract vector concretely as a list of coordinates once you fix a basis — and every basis of a given (finite-dimensional) vector space has the same number of elements, a fact that lets dimension be defined as a property of \(V\) itself, not of any particular basis you happened to choose. Linear Algebra is where this machinery gets used in earnest.

11.5 What a vector space is NOT

Not \(\mathbb{R}^n\) — \(\mathbb{R}^n\) is one example among many, not the definition, as the four motivating examples above were built to show. Not inherently equipped with a length or an angle — nothing in the five axioms above mentions how “long” a vector is or what it means for two vectors to be “perpendicular.” Those require additional structure layered on top, which is exactly what Inner Product Spaces adds next. A bare vector space has addition and scaling and nothing else; it’s easy to unconsciously import geometric intuition (angles, lengths) that the axioms haven’t actually earned yet.

11.6 If you’ve done linear algebra

You already know \(A\vec{x} = \vec{b}\). What the axioms above make precise is what \(\vec{x}\) is: an element of a vector space, nothing more specific required. When \(\vec{x}\) happens to be a tuple in \(\mathbb{R}^n\), that’s a choice of representation, not a fact forced by the definition — the same underlying idea works with \(\vec{x}\) ranging over polynomials or functions, which is exactly the setting Differential Equations and the functional analysis this site previews live in. Linear Algebra makes precise what \(A\) is doing to \(\vec{x}\) — spoiler: not “a grid of numbers,” but a transformation that a grid of numbers happens to represent once a basis is fixed.

11.7 Where this leads

A structure is only half the story without the maps between its instances. Linear Algebra is the theory of vector spaces and the linear maps connecting them — including the fact, central to everything from computer graphics to machine learning, that a matrix is just what a linear map looks like once you’ve chosen coordinates. This site’s table of contents takes a short detour first, through Inner Product Spaces and Metric Spaces, to pick up the notions of angle and distance a bare vector space deliberately doesn’t have.