Building Towards Homology: Vector Spaces and Modules

One of the more advanced topics in topology that I'd like to get to is homology. Homology is a major topic that goes beyond just algebraic topology, and it's really very interesting. But to understand it, it's useful to have some understandings of some basics that I've never written about. In particular, homology
uses chains of modules. Modules, in turn, are a generalization of the idea of a vector space. I've said a little bit about vector spaces when I
was writing about the gluing axiom, but I wasn't complete or formal in my description of them. (Not to mention the amount of confusion that I caused by sloppy writing
in those posts!) So I think it's a good idea to cover the idea in a fresh setting here.

So, what's a vector space? It's yet another kind of abstract algebra. In this case, it's an algebra built on top of a field (like the real numbers), where the values are a set of objects where there are two operations: addition of two vectors, and scaling a vector by a value from the field.

To define a vector space, we start by taking something like the real numbers: a set whose values form a field. We'll call that basic field F, and the elements of F we'll call scalars. We can then define a vector space over F as a set V whose members are called vectors, and which has two operations:

Vector Addition

An operation mapping two vectors to a third vector, +:V×V→V

Scalar Multiplication

An operation mapping a scalar and a vector to another scalar: *:F×V→V

Vector addition forms an abelian group over V, and scalar multiplication is distributive over vector addition and multiplication in the scalar field. To be complete, this means that the following properties hold:

• (V,+) are an Abelian group
  • Vector addition is associative: ∀a,b,c∈V: a+(b+c)=(a+b)+c
  • Vector addition has an identity element, 0; ∀a∈V:a+0=0+a=a.
  • Vector addition has an inverse element: ∀a∈V:(∃b∈V:a+b=0.) The additive inverse of
    a vector a is normally written -a. (Up to this point, this defines (V,+) is a group.)
  • Vector addition is commutative: ∀a,b∈V: a+b=b+a. (The addition of this commutative rule is what makes it an abelian group.)
• Scalar Multiplication is Distributive
  • Scalar multiplication is distributive over vector addition: ∀a∈F,∀b,c∈V, a*(b+c)=a*b+a*c
  • Scalar multiplication is distributive over addition in F: ∀a,b∈F,∀c∈V: (a+b)*c = (a*c) + (b*c).
  • Scalar multiplication is associative with multiplication in F: ∀a,b∈F,c∈V: (a*b)*c = a*(b*c).
  • The multiplicative identity for multiplication in F is also
    the identity element for scalar multiplication: ∀a∈V: 1*a=a.

So what does all of this mean? It really means that a vector space is a structure
over a field where the elements can be added (vector addition) or scaled (scalar
multiplication). Hey, isn't that exactly what I said at the beginning?

One obvious example of a vector space is a Euclidean space. Vectors are arrows from
the origin to some point in the space - and so they can be represented as ordered
tuples. So for example, ℜ³ is the three-dimensional euclidean space; points (x,y,z) are vectors. Adding two vectors (a,b,c)+(d,e,f)=(a+d,b+e,c+f); and scalar multiplication x*(a,b,c)=(x*a,x*b,x*c).

Following the same basic idea as the euclidean spaces, we can generalize to matrices of a particular size, each of which is a vector space. There are also ways of creating vector spaces using polynomials, various kinds of functions, differential equations, etc.

In homology, we'll actually be interested in modules. A module is just a
generalization of the idea of a vector space. But instead of using a field as a basis the way that you do in a vector space, in a module, the basis is just a general ring; so the basis is less constrained: a field is a commutative ring with multiplicative inverses of all values except 0, and distinct additive and multiplicative identities. So a module does not require multiplicative inverse for the scalars; nor does it require scalar multiplication to be commutative.