vector space

vector space
an additive group in which addition is commutative and with which is associated a field of scalars, as the field of real numbers, such that the product of a scalar and an element of the group or a vector is defined, the product of two scalars times a vector is associative, one times a vector is the vector, and two distributive laws hold. Also called linear space.

* * *

In mathematics, a collection of objects called vectors, together with a field of objects (see field theory), known as scalars, that satisfy certain properties.

The properties that must be satisfied are: (1) the set of vectors is closed under vector addition; (2) multiplication of a vector by a scalar produces a vector in the set; (3) the associative law holds for vector addition, u + (v + w) = (u + v) + w; (4) the commutative law holds for vector addition, u + v = v + u; (5) there is a 0 vector such that v + 0 = v; (6) every vector has an additive inverse (see inverse function), v + (-v) = 0; (7) the distributive law holds for scalar multiplication over vector addition, n(u + v) = nu + nv; (8) the distributive law also holds for vector multiplication over scalar addition, (m + n)v = mv + nv; (9) the associative law holds for scalar multiplication with a vector, (mn)v = m(nv); and (10) there exists a unit vector 1 such that 1v = v. The set of all polynomials in one variable with real coefficients is an example of a vector space.

* * *

      a set of multidimensional quantities, known as vectors (vector), together with a set of one-dimensional quantities, known as scalars, such that vectors can be added together and vectors can be multiplied by scalars while preserving the ordinary arithmetic properties (associativity, commutativity, distributivity, and so forth). Vector spaces are fundamental to linear algebra (algebra, linear) and appear throughout mathematics and physics.

      The idea of a vector space developed from the notion of ordinary two- and three-dimensional spaces as collections of vectors {u, v, w, …} with an associated field of real numbers (real number) {abc, …}. Vector spaces as abstract algebraic entities were first defined by the Italian mathematician Giuseppe Peano (Peano, Giuseppe) in 1888. Peano called his vector spaces “linear systems” because he correctly saw that one can obtain any vector in the space from a linear combination of finitely many vectors and scalars—av + bw + … + cz. A set of vectors that can generate every vector in the space through such linear combinations is known as a spanning set. The dimension of a vector space is the number of vectors in the smallest spanning set. (For example, the unit vector in the x-direction together with the unit vector in the y-direction suffice to generate any vector in the two-dimensional Euclidean plane when combined with the real numbers.)

      The linearity of vector spaces has made these abstract objects important in diverse areas such as statistics, physics, and economics, where the vectors may indicate probabilities, forces, or investment strategies and where the vector space includes all allowable states.

* * *

Universalium. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… …   Wikipedia

  • vector space — noun a) A type of set of vectors that satisfies a specific group of constraints. A vector space is a set of vectors which can be linearly combined. b) A set V, whose elements are called vectors , together with a binary operation + forming a… …   Wiktionary

  • vector space — vektorių erdvė statusas T sritis radioelektronika atitikmenys: angl. vector space vok. Vektorraum, m rus. векторное пространство, n; пространство векторов, n pranc. espace vectoriel, m …   Radioelektronikos terminų žodynas

  • vector space — vektorinė erdvė statusas T sritis fizika atitikmenys: angl. vector space vok. Vektorraum, m rus. векторное пространство, n pranc. espace vectoriel, m …   Fizikos terminų žodynas

  • Vector space model — (or term vector model ) is an algebraic model for representing text documents (and any objects, in general) as vectors of identifiers, such as, for example, index terms. It is used in information filtering, information retrieval, indexing and… …   Wikipedia

  • vector space — noun Date: 1937 a set of vectors along with operations of addition and multiplication such that the set is a commutative group under addition, it includes a multiplicative inverse, and multiplication by scalars is both associative and… …   New Collegiate Dictionary

  • vector space — noun : a set representing a generalization of a system of vectors and consisting of elements which comprise a commutative group under addition, each of which is left unchanged under multiplication by the multiplicative identity of a field, and… …   Useful english dictionary

  • Topological vector space — In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a… …   Wikipedia

  • Normed vector space — In mathematics, with 2 or 3 dimensional vectors with real valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of vector length are crucial. 1. The zero… …   Wikipedia

  • Locally convex topological vector space — In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) which generalize normed spaces. They can be defined as topological vector… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”