for courses in advanced linear algebra, this top selling theorem proof text presents a careful treatment of the principal topics of linear algebra and illustrates the power of the subject through a variety of application.it emphasizes the symbiotic relationship between linear transformation and matrices, but states theorems in the more general infinite dimensional case where appropriate.
The primary purpose of this fourth edition of Linear Algebra is to present
a careful treatment of the principal topics of linear algebra and to illustrate
the power of the subject through a variety of applications. Our major thrust
emphasizes the symbiotic relationship between linear transformations and
matrices. However, where appropriate, theorems are stated in the more general infinite-dimensional case. For example, this theory is applied to finding
solutions to a homogeneous linear differential equation and the best approximation by a trigonometric polynomial to a continuous function.
Although the only formal prerequisite for this book is a one-year course
in calculus, it requires the mathematical sophistication of typical junior and
senior mathematics majors. This book is especially suited for a second course
in linear algebra that emphasizes abstract vector spaces, although it can be
used in a first course with a strong theoretical emphasis.
The book is organized to permit a number of different courses (ranging
from three to eight semester hours in length) to be taught from it. The
core material (vector spaces, linear transformations and matrices, systems of
linear equations, determinants, diagonalization, and inner product spaces) is
found in Chapters 1 through 5 and Sections 6.1 through 6.5. Chapters 6 and
7, on inner product spaces and canonical forms, are completely independent
and may be studied in either order. In addition, throughout the book are
applications to such areas as differential equations, economics, geometry, and
physics. These applications are not central to the mathematical development,
however, and may be excluded at the discretion of the instructor.
We have attempted to make it possible for many of the important topics
of linear algebra to be covered in a one-semester course. This goal has led
us to develop the major topics with fewer preliminaries than in a traditional
approach. (Our treatment of the Jordan canonical form, for instance, does
not require any theory of polynomials.) The resulting economy permits us to
cover the core material of the book (omitting many of the optional sections
and a detailed discussion of determinants) in a one-semester four-hour course
for students who have had some prior exposure to linear algebra.
