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AppendixBFields and Vector Spaces

In this Appendix we give a brief introduction to the axiomatic theory of fields, which underlies the notion of abstract vector spaces.

We first give the modern algebraic definition of fields, of which the real and complex numbers are the main examples.

Definition

A commutative ring is tuple (Λ,+,·,−,0,1) consisting of a set Λ, two binary operations + and · of type Λ×Λ→Λ, a unary operation −:Λ→Λ, and two constants 0,1∈Λ, satisfying the following laws, for all λ,µ,ν∈Λ:

(λ+µ)+ν=λ+(µ+ν)(a1)λ+0=λ(a2)λ+µ=µ+λ(a3)λ+(−λ)=0(a4)(λµ)ν=λ(µν)(m1)λ1=λ(m2)λµ=µλ(m3)λ(µ+ν)=λµ+λν(am)

A field is a commutative ring in which every nonzero element λ has an inverse, i.e., an element λ−1 satisfying λλ−1=λ−1λ=1.

The operation + is called the addition, the operation · is called the multiplication, and the operation − is called the negation operation of the field. As usual, we leave out the dot when writing multiplications. To be super precise, we would have to decorate the operations and the constants with Λ to disambiguate them from the usual ones, as in (Λ,+Λ,·Λ,−Λ,0Λ,1Λ). The laws (a1) and (m1) are called the associative laws, the laws (a2) and (m2) concern the neutral elements 0 and 1, the laws (a3) and (m3) are called the commutative laws, and the law (am) is called the distributive law.

As a consequence of the laws, we can prove other identities, such as 1λ=λ, (λ+µ)ν=λν+µν, and 0λ=0, for all λ,µ,ν∈Λ. For instance, the last one follows from the calculation

0λ=(0+0)λ=0λ+0λ,

together with (a4).

It is possible to consider many variations of these axioms, to give possibly non-commutative rings, rings without units (1 ), skew-fields, etc. But all the fields laws are familiar from real number arithmetic in R, and they also hold for complex numbers C (Appendix A), as the reader is invited to check. The point of introducing the notion of a field is that all the theorems of linear algebra hold over any field and that there are many other examples:

  • The rational numbers Q: the reals numbers of the form p/q where qA=0.
  • More general number fields Q⊆Λ⊆C, for example Q(B2), the smallest subfield of C containing a square root of 2, or the smallest subfield of C containing a solution to some polynomial equation with rational coefficients, etc. (This way leads to Galois theory.)
  • Finite fields Fpn with pn elements, where p is a prime number. As an example, we can describe a field F4 with 4 elements 0,1,a,a−1, where 1+1=0 and a2=a+1.
  • The field of rational functions Λ(X) over a field Λ consists of rational functions, i.e., those of the form P/Q, where P,Q are polynomials with coefficients from Λ and QA=0.
  • etc.

We can now give the definition of abstract vector spaces.

Definition

A vector space over the field Λ is a tuple (V,+,·,−,0) consisting of a set V, binary operations +:V×V→V and ·:Λ×V→V, a unary operation −:V→V, and a constant 0∈V, satisfying the following laws, for all x,y,z∈V and λ,µ∈Λ:

(x+y)+z=x+(y+z)(a1)x+0=x(a2)x+y=y+x(a3)x+(−x)=0(a4)(λµ)x=λ(µx)(v1)λ(x+y)=λx+λy(v2)(λ+µ)x=λx+µx(v3)1x=x(v4)

The operation + is again called the addition (of vectrs), but the operation · is called the scalar multiplication; again, it is usually left out of notations. These operations allow us to produce linear combinations in a meaningful way, and we can prove all the theorems of linear algebra for these, at least when they are finite dimensional, i.e., admit a finite ordered basis, as defined in Section 3.4. The standard examples are the spaces Λn of column vectors of length n, for any n∈N. Other examples:

  • Vector subspaces (Section 3.3) of any vector space.
  • The set of polynomials
    Λ[X]={anXn+···+a1X+a0|ai∈Λ}
    with the obvious addition and scalar multiplications.
  • The set of m×n matrices over Λ (Section 2.4).
  • The set of linear maps f:V→W between two vector spaces over Λ (Section 4.3).
  • The quotient space V/U of a vector space V with respect to a subspace U: This is obtained a the quotient of V by the equivalence relation ∼ defined by x∼y if and only if y−x∈U. The operations are inhered from V in the sense that λ[x]=[λx] and [x]+[y]=[x+y] are well defined.
  • etc.

To read more about abstract algebra, including the theory of commutative rings and fields, and linear algebra over fields, see most undergraduate algebra textbooks, e.g., A Survey of Modern Algebra by Birkhoff and Mac Lane.