This book is concisely written to provide comprehensive core materials for a year-long course in Linear Algebra for senior undergraduate and beginning graduate students in mathematics, science, and engineering. Students who gain profound understanding and grasp of the concepts and methods of this course will acquire an essential knowledge foundation to excel in their future academic endeavors.

Throughout the book, methods and ideas of analysis are greatly emphasized and used, along with those of algebra, wherever appropriate, and a delicate balance is cast between abstract formulation and practical origins of various subject matters.

The book is divided into nine chapters. The first seven chapters embody a traditional course curriculum. An outline of the contents of these chapters is sketched as follows.

In Chapter 1 we cover basic facts and properties of vector spaces. These include definitions of vector spaces and subspaces, concepts of linear dependence, bases, coordinates, dimensionality, dual spaces and dual bases, quotient spaces, normed spaces, and the equivalence of the norms of a finite-dimensional normed space.

In Chapter 2 we cover linear mappings between vector spaces. We start from the definition of linear mappings and discuss how linear mappings may be concretely represented by matrices with respect to given bases. We then introduce the notion of adjoint mappings and quotient mappings. Linear mappings from a vector space into itself comprise a special but important family of mappings and are given a separate treatment later in this chapter. Topics studied there include invariance and reducibility, eigenvalues and eigenvectors, projections, nilpotent mappings, and polynomials of linear mappings. We end the chapter with a discussion of the concept of the norms of linear mappings and use it to show that being invertible is a generic property of a linear mapping and then to show how the exponential of a linear mapping may be constructed and understood.