Vector 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, &math.n;(u + v) = &math.n;u + &math.n;v; (8) the distributive law also holds for vector multiplication over scalar addition, (&math.m; + &math.n;)v = &math.m;v + &math.n;v; (9) the associative law holds for scalar multiplication with a vector, (&math.m;&math.n;)v = &math.m;(&math.n;v); 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.