Vector Spaces

This chapter focuses on vector spaces over the real numbers. The zero vector space is the vector space consisting only of the zero vector. Most of the results on vector spaces either apply directly to the zero vector space or can be made applicable with a minor reworking of definitions and proofs. Following the usual convention in differential geometry, people index the scalars and vectors in a linear combination with superscripts and subscripts, respectively. This opens the door to the Einstein summation convention. The chapter explains that a vector space is finite-dimensional if it has a finite unordered basis. Finite-dimensional vector spaces have an associated invariant that largely characterizes them. The chapter defines the dual (vector) space of a vector space. From this humble beginning, the theory of differential forms will eventually emerge. Continuing with that theme, the chapter associates with a corresponding linear map between their dual spaces.