A vector space is a set of objects (called vectors) that can be added together and multiplied by numbers (called scalars), obeying a fixed list of rules. Vector spaces are defined over a field, and they are the setting for linear algebra โ one of the most widely applied branches of mathematics, from computer graphics to quantum mechanics.
Think of an arrow pointing from the origin to a point on a map โ that's a vector. You can add two arrows together (place one at the tip of the other, and draw a new arrow from start to finish). You can also stretch or shrink an arrow by multiplying it by a number โ double it, halve it, flip it around by multiplying by โ1.
The surprising part: vector spaces don't have to be arrows in space. A list of 5 numbers (like a spreadsheet row) is a vector. A polynomial like 3xยฒ + 2x + 1 is a vector, in a space of polynomials. Even the set of all possible sound waves can be treated as a vector space. Anything that can be added and scaled sensibly qualifies.
A vector space V over a field F is a set equipped with two operations โ vector addition (V ร V โ V) and scalar multiplication (F ร V โ V) โ satisfying 8 axioms: associativity and commutativity of addition, existence of a zero vector, existence of additive inverses, and four axioms governing how scalar multiplication interacts with addition (distributive laws, compatibility with field multiplication, and identity: 1ยทv = v).
| Vector space | Field | Elements |
|---|---|---|
| โโฟ | โ | n-tuples of real numbers |
| โโฟ | โ | n-tuples of complex numbers |
| Mโโโ(โ) | โ | mรn real matrices |
| P(x) | โ | Polynomials with real coefficients |
| C[a,b] | โ | Continuous functions on [a,b] |
A basis is a set of vectors that are linearly independent (none is a combination of the others) and span the whole space (every vector can be written as a combination of them). Every vector space has a basis, and all bases of a given space have the same size โ called the dimension. โยณ has dimension 3; its standard basis is {(1,0,0), (0,1,0), (0,0,1)}.
A linear transformation T : V โ W between vector spaces satisfies T(u+v) = T(u)+T(v) and T(cv) = cT(v). Every linear transformation between finite-dimensional spaces can be represented by a matrix once bases are chosen โ this is why matrices are so central to linear algebra.
A subset W โ V is a subspace if it is itself a vector space under the same operations โ equivalently, if it's closed under addition and scalar multiplication and contains the zero vector. The set of solutions to a homogeneous linear equation Ax = 0 is always a subspace, called the kernel or null space of A.
Vector spaces need not be finite-dimensional. The space of all polynomials has infinite dimension (no finite basis can span it, since polynomials of arbitrarily high degree exist). Function spaces like C[a,b] (continuous functions) or Lยฒ[a,b] (square-integrable functions) are infinite-dimensional and are studied in functional analysis. These spaces require additional structure (norms, inner products, completeness) to do useful analysis โ leading to Banach spaces and Hilbert spaces.
An inner product โจu,vโฉ on a vector space adds a notion of angle and length: โvโ = โโจv,vโฉ. Two vectors are orthogonal if โจu,vโฉ = 0. The Pythagorean theorem generalises: if u โฅ v, then โu+vโยฒ = โuโยฒ + โvโยฒ. A Hilbert space is a complete inner product space โ the setting for quantum mechanics, where physical states are vectors and observables are linear operators.
For a linear map T : V โ W with V finite-dimensional: dim(V) = dim(ker T) + dim(im T). This fundamental theorem relates the dimension of the domain to the dimensions of the kernel (what's sent to zero) and the image (what's actually reached) โ a cornerstone result used throughout linear algebra.
The state of a quantum system is represented as a vector in a complex Hilbert space. Physical observables correspond to linear operators (Hermitian operators) on this space, and possible measurement outcomes correspond to eigenvalues of these operators. Dirac's bra-ket notation (โจฯ|, |ฯโฉ) is vector space notation specifically adapted for quantum theory. This application illustrates how an apparently abstract algebraic structure became indispensable to fundamental physics.
The abstract axiomatic definition of a vector space was given by Giuseppe Peano (1888), extending earlier geometric work by Hermann Grassmann (1844) on "extension theory," which was largely ignored in its time due to its unusual notation and abstract presentation. The theory was popularised and connected to matrix theory over the following decades, becoming standard in the 20th century.