Vector Space
A vector space is a set of objects called vectors, together with two operations — vector addition and scalar multiplication — that satisfy eight axioms including closure, associativity, distributivity, and the existence of zero and inverse elements. Vector spaces provide the foundational framework for linear algebra, generalising the familiar geometry of arrows in 2D and 3D to abstract settings of any dimension. They are essential in physics, engineering, computer graphics, machine learning, and quantum mechanics, where states, signals, and transformations are all described as elements of appropriate vector spaces.
Eight Axioms of a Vector Space
| Axiom | Symbol | Description |
|---|---|---|
| Closure under addition | u + v ∈ V | Sum of two vectors is still in the space |
| Commutativity | u + v = v + u | Order of addition does not matter |
| Associativity | (u+v)+w = u+(v+w) | Grouping of addition does not matter |
| Zero vector | v + 0 = v | Additive identity exists |
| Additive inverse | v + (−v) = 0 | Every vector has a negative |
| Scalar distributivity | a(u+v) = au + av | Scalars distribute over vector sum |
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Related Terms
Basis (Linear Algebra)
A basis of a vector space is a set of linearly independent vectors that spans the entire space, meaning every vector in the space can be expressed as a unique linear combination of the basis vectors. The number of vectors in any basis of a finite-dimensional vector space is the same and equals the dimension of the space. Bases are fundamental in coordinate systems, data compression, Fourier analysis, and the numerical solution of linear systems.
Linear Independence
A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the others, equivalently if the only solution to the equation c₁v₁ + c₂v₂ + … + cₙvₙ = 0 is the trivial solution where all scalars cᵢ equal zero. If at least one non-zero scalar solution exists, the set is called linearly dependent. Linear independence is the key criterion for determining whether a set of vectors forms a basis, and it underlies the concepts of dimension, rank, and solutions to linear systems.
Matrix (Linear Algebra)
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, where an m×n matrix has m rows and n columns. Matrices represent linear transformations between vector spaces, encode systems of linear equations, and serve as the primary computational tool in linear algebra. They are indispensable across engineering, physics, computer science, statistics, and machine learning, where they model data, rotations, reflections, and the weights of neural networks.
From Latin "vector" (carrier, from "vehere" — to carry) and "spatium" (space). The abstract algebraic definition was formalised by Giuseppe Peano in 1888 in his work "Calcolo Geometrico".