# vector space

## 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, 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.

This entry comes from Encyclopædia Britannica Concise.
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