Vector space

In this article, vectors are represented in boldface to distinguish them from scalars.^{[nb 1][1]}


A vector space over a field F is a non-empty set V together with a binary operation and a binary function that satisfy the eight axioms listed below. In this context, the elements of V are commonly called vectors, and the elements of F are called scalars.^[2]


To have a vector space, the eight following axioms must be satisfied for every u, v and w in V, and a and b in F.^[3]


When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space.^[4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered. Such a vector space is called an F-vector space or a vector space over F.^[5]


An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the field F into the endomorphism ring of this group.^[6]


Subtraction of two vectors can be defined as $${\displaystyle \mathbf {v} -\mathbf {w} =\mathbf {v} +(-\mathbf {w} ).}$$


Direct consequences of the axioms include that, for every ${\displaystyle s\in F}$ and ${\displaystyle \mathbf {v} \in V,}$ one has


Even more concisely, a vector space is a module over a field.^[7]

History[edit]

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, French mathematicians René Descartes and Pierre de Fermat founded analytic geometry by identifying solutions to an equation of two variables with points on a plane curve.^[18] To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines, and planes, which are predecessors of vectors.^[19] Möbius (1827) introduced the notion of barycentric coordinates.^[20] Bellavitis (1833) introduced an equivalence relation on directed line segments that share the same length and direction which he called equipollence.^[21] A Euclidean vector is then an equivalence class of that relation.^[22]


Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions by the latter.^[23] They are elements in R² and R⁴; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations.


In 1857, Cayley introduced the matrix notation which allows for harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.^[24] In his work, the concepts of linear independence and dimension, as well as scalar products are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called algebras. Italian mathematician Peano was the first to give the modern definition of vector spaces and linear maps in 1888,^[25] although he called them "linear systems".^[26] Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897, Salvatore Pincherle adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.^[27]


An important development of vector spaces is due to the construction of function spaces by Henri Lebesgue. This was later formalized by Banach and Hilbert, around 1920.^[28] At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.^[29]