a set of multidimensional quantities, known as vectors (vector), together with a set of one-dimensional quantities, known as scalars, such that vectors can be added together and vectors can be multiplied by scalars while preserving the ordinary arithmetic properties (associativity, commutativity, distributivity, and so forth). Vector spaces are fundamental to linear algebra (algebra, linear) and appear throughout mathematics and physics.

The idea of a vector space developed from the notion of ordinary two- and three-dimensional spaces as collections of vectors {u, v, w, …} with an associated field of real numbers (real number) {a,b,c,…}. Vector spaces as abstract algebraic entities were first defined by the Italian mathematician Giuseppe Peano (Peano, Giuseppe) in 1888. Peano called his vector spaces “linear systems” because he correctly saw that one can obtain any vector in the space from a linear combination of finitely many vectors and scalars—av+bw+…+cz. A set of vectors that can generate every vector in the space through such linear combinations is known as a spanning set. The dimension of a vector space is the number of vectors in the smallest spanning set. (For example, the unit vector in the x-direction together with the unit vector in the y-direction suffice to generate any vector in the two-dimensional Euclidean plane when combined with the real numbers.)

The linearity of vector spaces has made these abstract objects important in diverse areas such as statistics, physics, and economics, where the vectors may indicate probabilities, forces, or investment strategies and where the vector space includes all allowable states.