Vector Spaces and Bases

Abstract

Vector spaces and their ancillary structures provide the common language of linear algebra, and, as such, are an essential prerequisite for understanding contemporary applied (and pure) mathematics. The key concepts of vector space, subspace, linear independence, span, and basis will appear, not only in linear systems of algebraic equations and the geometry of n-dimensional Euclidean space, but also in the analysis of linear differential equations, linear boundary value problems, Fourier analysis, signal processing, numerical methods, and many, many other fields. Therefore, in order to master modern linear algebra and its applications, the first order of business is to acquire a firm understanding of fundamental vector space constructions.