A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field.
Formalized by Giuseppe Peano in 1888, though roots trace back to analytic geometry.
Fundamental to linear algebra, forming the basis for many areas of mathematics, physics, and engineering.