Overview

This appendix is divided into twelve sections. Among them, Secs. B.1-B.6 are devoted to static structures of vector spaces themselves, while Secs.. B.7-B.12 are mainly concerned with dynamic relations between vector spaces, namely the study of linear transformations. Most topics are stated in the realm of finite dimensional vector spaces. From Sec. B.4 on, some exercise problems are attached as parts of the contents. Few geometric interpretations of abstract results are touched and the methods adopted are almost purely algebraic. Essentially no proofs are given. In short, the manner presented is mostly adopted in the nowadays linear algebra books.

A vector space V over a field is defined axiomatically in Sec. B.1, with , and M( m, n; ) as concrete examples along with subspace operations. Via the techniques of linear combination, dependence and independence introduced in Sec. B.2, Sec. B.3 introduces the basis and dimension for a vector space. The concept of matrices over a field and their algebraic operations are in Sec. B.4, and the elementary row or column operations on a matrix are in Sec. B.5. The determinant function on square matrices is sketched in Sec. B.6.

Section B.7 is devoted to linear transformation (functional, operator, or isomorphism) and its matrix representation with respect to bases. Section B.8 investigates a matrix and its transpose, mostly from the viewpoint of linear transformations. Inner product spaces with specified linear operators on them such as orthogonal, normal, etc. are in Sec. B.9. Eigenvalues and...