A main mathematical structure in linear algebra. A vector space V over a field F consists of a set V with operations of addition +: V × V → V, and scalar multiplication F × V → V, such that V is an abelian group under + and further

where α‎,β‎ ∈ F and v,w ε‎ V. Elements of F are called scalars and elements of V are called vectors. Examples of real vectors spaces (where F = ℝ) include:

• the space of m × n real matrices;

• the space of polynomials with real coefficients;

• the solution space of homogeneous simultaneous linear equations;

• the kernel and image of a linear map;

• the space Hom(V,W) of all linear maps between vector spaces V and W;

• sequence spaces (see c and l^∞);

• the space of continuous real-valued functions on a topological space.

A vector space V is finite-dimensional if it has a finite basis, in which case all bases will contain the same number of elements, the space’s dimension. Once a basis is chosen, coordinates may be uniquely assigned to vectors identifying V with ℝ^dimV. See affine space, dual space, inner product, module, normed vector space, n-dimensional space, vector subspace.