Section 13 : Modules

When you learn vector spaces at Imperial College London, you start off with the theory of real vector spaces, on the basis that this is “more concrete” or something. You do examples in two and three dimensions and talk about vectors and matrices. Multiplication of matrices is explained via some weird formula.

Later on, you begin to abstract away from the concrete model \(\mathbb{R}^3\) all of whose elements have a unique “formula” like \((1,2,3)\), and move on to the general space with a basis \(\mathbb{R}^n\), and finally to the general real vector space \(V\), an abstraction of the concrete axioms satisfied by \(mathbb{R}^2\) and \(mathbb{R}^3\). You can still add two vectors: \(v_1+v_2 \in V\), and multiply a vector by a scalar \(r\boldsymbol{\cdot} v\in V\) if \(r\in R\) and \(v \in V\), but there is no longer a “concrete model” – an lement of V has now become an opaque abstraction; the elements no longer have “formulas” like \((\sin(\theta),\cos(\theta),37)\) and if you want them back (for example if you want to do an explicit computation like they do in applied maths) then you need to pick a basis.