# 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*) = *n**u* + *n**v*; (8) the distributive law also holds for vector multiplication over scalar addition, (*m* + *n*)*v* =* mv* +

*; (9) the associative law holds for scalar multiplication with a vector, (*

*n*v*m*

*n*)

*v*=

*m*(

*); and (10) there exists a unit vector*

*n*v*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*.

For the full entry on vector space, visit Britannica.com.

## Learn More About

**Ask The Editor** Videos

## Seen & Heard

What made you look up *vector space*? Please tell us what you were reading, watching or discussing that led you here.