vector space

vector space


Definition of VECTOR SPACE

:  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 distributive

First Known Use of VECTOR SPACE


vector space

noun    (Concise Encyclopedia)

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.


Next Word in the Dictionary: vector sum
Previous Word in the Dictionary: vector psychology
All Words Near: vector space

Seen & Heard

What made you want to look up vector space? Please tell us where you read or heard it (including the quote, if possible).

Get Our Free Apps
Voice Search, Favorites,
Word of the Day, and More
Join Us on FB & Twitter
Get the Word of the Day and More