Vector space

The fundamental concept in linear algebra is that of a vector space or linear space. It is a generalization of the set of all geometrical vectors and is used throughout modern mathematics.

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:

1. v+w belongs to V.
   V is closed under vector addition.
2. u+(v+w)= (u+v)+w.
   Associativity of vector addition in V.
3. There exists a neutral element 0 in V, such that for all elements v in V, v+0=v.
   Existence of an additive identity element in V.
4. For all v in V, there exists an element -v in V, such that v+(-v)=0.
   Existence of additive inverses in V.
5. v+w=w+v.
   Commutativity of vector addition in V.
6. a*v belongs to V.
   V is closed under scalar multiplication.
7. a*(b*v)=(a*b)*v.
   Associativity of scalar multiplication in V.
8. If 1 denotes the multiplicative identity of the field F, then 1*v=v.
   Neutrality of one.
9. a*(v+w)=a*v+a*w.
   Distributivity with respect to vector addition.
10. (a+b)*v=a*v+b*v.
    Distributivity with respect to field addition.

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:

a*0 = 0*v = 0

-(a*v) = (-a)*v = a*(-v)

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.

Terminology

• A vector space over R, the set of real numbers, is called a real vector space.
• A vector space over C, the set of complex numbers, is called a complex vector space.
• A vector space with a defined distance concept, i.e. a norm, is called a normed vector space.

Examples

• Example 1: The vector space Rⁿ, over R, with component-wise operations
  • More generally, Fⁿ, over F, with component-wise operations
• Example 2: The set of (mxn) matrices with complex elements over C
  • More generally, the set of (mxn) matrices over an arbitrary field F
• Example 3: The set of all continuous real-valued functions on a closed interval
• Given a vector space V over F, and some set X, then the set of all functions X -> V forms a vector space over F
• F[x]: The set of all ploynomials with coefficents out of F, over F.
• The finite field GF(pⁿ), over GF(p)
• C, over R
• R, over Q (the rational numbers)

Subspaces and bases

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.

Linear maps

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.

Generalization

In abstract algebra, the concept of a vector space is generalized to modules by replacing the underlying field F by a ring and retaining the above 10 axioms.