Matrices and Linear Transformations

Second Edition

Charles G. Cullen

• Éditeur: Dover Publications
• Année: 2012
• ISBN:
  9780486132419
• Édition: 2
• Sujet: Mathematics
• Sous-sujet: Algebra

`Comprehensive . . . an excellent introduction to the subject.` — Electronic Engineer's Design Magazine.
This introductory textbook, aimed at sophomore- and junior-level undergraduates in mathematics, engineering, and the physical sciences, offers a smooth, in-depth treatment of linear algebra and matrix theory. The major objects of study are matrices over an arbitrary field.
Contents include Matrices and Linear Systems; Vector Spaces; Determinants; Linear Transformations; Similarity: Part I and Part II; Polynomials and Polynomial Matrices; Matrix Analysis; and Numerical Methods.
The first seven chapters, which require only a first course in calculus and analytic geometry, deal with matrices and linear systems, vector spaces, determinants, linear transformations, similarity, polynomials, and polynomial matrices. Chapters 8 and 9, parts of which require the student to have completed the normal course sequence in calculus and differential equations, provide introductions to matrix analysis and numerical linear algebra, respectively. Among the key features are coverage of spectral decomposition, the Jordan canonical form, the solution of the matrix equation AX = XB, and over 375 problems, many with answers.

1

Matrices
and
Linear
Systems

1.1 INTRODUCTION

We will begin by discussing two familiar problems which will serve as motivation for much of what will follow.

First of all, you are all familiar with the problem of finding the solution (or solutions) of a system of linear equations. For example the system

can be easily shown to have the unique solution x = 1, y = – 1, z = 1. Most of the techniques you have learned for finding solutions of systems of this type become very unwieldy if the number of unknowns is large or if the coefficients are not integers. It is not uncommon today for scientists to encounter systems like (1.1) containing several thousand equations in several thousand unknowns. Even using the most efficient techniques known, a fantastic amount of arithmetic must be done to solve such a system. The development of high-speed computational machines in the last 20 years has made the solution of such problems feasible.

Using the elementary analytical geometry of three-space, one can provide a convenient and fruitful geometric interpretation for the system (1.1). Since each of the three equations represents a plane, we would normally expect the three planes to intersect in precisely one point, in this case the point with coordinates (1, – 1, 1). Our geometric point of view suggests that this will not always be the case for such systems since the following two special cases might arise:

1. Two of the planes might be parallel, in which case there would be no points common to the three planes and hence no solution of the system.

2. The planes might intersect in a line and hence there would be an infinite number of solutions.

The first of these special cases could be illustrated in the system obtained from (1.1) by replacing the third equation by 3x + 2y – z = 5; the second special case could be illustrated by replacing the third equation in (1.1) by the equation 4x + y = 3.

Let us now look at a general system like (1.1). Consider

where the a_ij and the k_i are known constants and the x_i are the unknowns, so that we have m equations in n unknowns. Note, by the way, the advantages of the double subscript notation in writing the general linear system (1.2). It may be difficult for you to interpret this s...