in mathematics, a set whose elements are generalized vectors and which is a commutative group under addition that is also closed under an operation of multiplication by elements of a given field having the properties that c (A + B) = cA + cB and (c + d) A = cA + dA, (cd) A = c (dA), and 1A = A where A, B are vectors, c, d are elements of the field, and 1 is the identity element of the field under multiplication