Vector space

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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.

Definition: A set V is a vector space over a field F, 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 an 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)

for all a in F and v in 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.

Examples:

  • /Example I: The vector space Rn, over R, with component-wise operations
    • More generally, Fn, over F, with component-wise operations
  • /Example II: The set of (mxn) matrices with complex elements over C
    • More generally, the set of (mxn) matrices over an arbitrary field F
  • /Example III: 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
  • The finite field GF(pn), over GF(p)
  • C, over R
  • R, over Q (the rational numbers)

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.

A morphism from a vector space V to a vector space W (necessarily over the same field) is called a linear transformation or "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.


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