Finitely Generated Vector Spaces

But before starting on this, let me try to explain again, in a different way, our approach. The one habit I’ve been trying to wean you of is the an over-reliance upon concrete examples to develop your understanding. The vector space R is a very concrete and familar example of a vector space over a field. To do calculations in this setting all you need to do is apply arithmetic (over and over and over). On the other hand, there are a number of other sets can be endowed with operations of scalar multiplication and vector addition so that they behave like R. So we have a certain dichotomy here; a concrete and familar object, R, and an associated set of patterns (the axioms of a vector space). What we are trying to do is deduce things from the patterns (axomatic vector space structure) that must be true for any object that satisfies the basic set of patterns. This allows us to say a whole lot about a whole lot of situations fairly succinctly.