A set V is a vector space over a field F (such as the field of real or of complex numbers, for example), if given an operation vector addition defined in V, denoted v+w for all v, w in V, and an operation scalar multiplication in V, denoted a*v for all v in V and a in F, the following 10 properties hold for all a, b in F and u, v, and w in V:
Properties 1 through 5 indicate that V is an abelian group under vector addition. Properties 6 through 10 apply to scalar multiplication of a vector v in V by a scalar a in F. (Note that Property 5 actually follows from the other 9.)
From the above properties, one can immediately prove the following handy formulas:
The members of a vector space are called vectors. The concept of a vector space is entirely abstract like the concepts of a group, ring, and field. To determine if a set V is a vector space one must specify the set V, a field F and define vector addition and scalar multiplication in V. Then if V satisfies the above 10 properties it is a vector space over the field F.
Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is called linearly independent. A linearly independent set whose span is the whole space is called a basis.
All bases for a given vector space have the same cardinality. Using Zorn's Lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the dimension of the vector space) representing the size of the basis. For instance the real vector spaces are just R0, R1, R2, R3, ..., R∞, ... As you would expect, the dimension of the real vector space R3 is three.
Given two vector spaces V and W over the same field, one can define linear transformations or "linear maps" from V to W. These are maps from V to W which are compatible with the relevant structure, i.e. they preserve sums and scalar products. The set of all linear maps from V to W is denoted L(V,W) and makes up a vector space over the same field. When bases for both V and W are given, linear maps can be expressed in terms of components as matrices.
An isomorphism is a linear map that is one-to-one and onto. If there exists an isomorphism between V and W, we call the two spaces isomorphic; they are then essentially identical.
The vector spaces over a fixed field F, together with the linear maps, form a category.