Chapter 2 - Vector Spaces and Linear Transformations

Abstract

This chapter presents the idea of a vector space, highlights few the consequences of the definition, and presents several examples. It explains the properties of vector spaces. Any vector space S has the following properties: (1) the zero vector 0 is unique, (2) for each U in S, −U is unique, (3) 0U = 0 for every U in S, (4) a0 = 0 for every scalar a, (5) (−1) U = −U, and (6) if cU = 0, then c = 0 or U = 0. The chapter focuses on subspaces of a vector space and several ideas that are important for the study of vector spaces, such as linear combination of vectors, linear dependence, and linear transformations. A linear transformation from a vector space S to a vector space T is a rule that assigns elements of S to elements of T in a way that preserves vector addition and scalar multiplication. A linear transformation that is one-to-one and onto is called an isomorphism. If L: S → T is an isomorphism, S is said to be isomorphic to T. The chapter also highlights eigenvalues and eigenvectors.