ABSTRACT
The goal of this chapter is to present a general axiomatic treatment of some fundamental concepts from linear algebra: linear independence, linear dependence, spanning sets, and bases. One benefit of the axiomatic approach is that it sweeps away a lot of irrelevant extra structure, isolating a few key properties that underlie the basic theorems about linear independence and bases. Even better, these axioms arise in other situations besides linear algebra. Hence, all the theorems deduced from the axioms will apply to those other situations as well. For example, we will prove the main theorems about the transcendence degree of field extensions by verifying the axioms of the general theory.
