In mathematics, a collection of objects called vector s, 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 = m v + n v ; (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 polynomial s in one variable with real coefficients is an example of a vector space.