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# Vector space

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One of the fundamental concepts in linear algebra is that of a vector space. For simplicity, we define a vector space over a field; a similar object defined over a ring is called a module.

Definition: A set V is a vector space over a field F, if given the operation vector addition defined in V, denoted v+w for all v,w in V, and the operation scalar multiplication in V, denoted a*(v) for all v in V and a in F, the following 10 properties hold:

1. For all v,w in V, v+w belongs to V.
   V is closed under vector addition.
2. For all u,v,w in V, u+(v+w)= (u+v)+w.
   Associativity of vector addition in V.
3. For all v in V, there exists an element 0 in V, such that 0+v=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 an additive inverse in V.
5. For all v,w in V, v+w=w+v.
   Commutativity of vector addition in V.
6. For all a in F and v in V, a*v belongs to V.
   V is closed under scalar multiplication.
7. For all a,b in F and v in V, a*(b*v)=(a*b)*v.
   Associativity of scalar multiplication in V.
8. For all v in V, there exists and element 1 in F, such that 1*v=v.
   Existence of an identity element for scalar multiplication in V.
9. For all v,w in V and a in F, a*(v+w)=a*v+a*w.
   Distributivity with respect ot vector addition.
10. For all a,b in F and v in V, (a+b)*v=a*v+a*w.
    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.

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.

/Example I: The vector space Rn, with component-wise operations
/Example II: The set of (mxn) matrices with complex elements
/Example III: The set of all real-valued functions on a closed interval

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 space have the same cardinality. If you accept the axiom of choice, then all vector spaces have bases, so vector spaces are fixed up to isomorphism by a single cardinal representing the size of the basis. For instance the real vector spaces are just R, R2, R3, ..., R∞, ... This is a very strong result, much stronger than equivalent results over more general modules.

A homomorphism from a vector space V to a vector space W (necessarily over the same field) is called a linear map. That is, a map is linear if and only if it preserves sums and scalar products. An isomorphism is a linear map that is one-to-one and onto. 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.

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Last edited August 16, 2001 3:12 pm (diff)
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