Linear Algebra

The following sections are included:

• Dedication
• Preface
• Contents

In this chapter we study algebraic properties of matrices, including operations on matrices, Gaussian elimination, and inverse matrices.

In this chapter, we present basic topics in linear algebra, like linear independence, vector subspace, basis and dimension, that will be needed in the rest of this book.

In this chapter we discuss some geometric aspects of linear algebra. As we will see, geometric interpretations of algebraic concepts provide useful insights that help us solve algebraic problems. On the other hand, algebraic methods often give us powerful tools for dealing with problems that are geometric in nature.

The following section is included:

• The determinant of a n × n matrix

Diagonal matrices play a crucial role in this chapter…

In this chapter we take another look at some familiar ideas from earlier chapters from a somewhat different point of view. While the new approach may seem less natural the first time we are exposed to it, it turns out to be quite useful when dealing with more advanced problems in linear algebra…

In Chapter 5 we studied matrices A that can be written in the form A = PDP⁻¹, where P is an invertible matrix and D is a diagonal matrix. Such a representation of a matrix can be very useful, but in applications we often encounter matrices that cannot be diagonalized in this way. In this chapter we present a method for representing matrices in a form similar to diagonalization that can be used for matrices that are not diagonalizable. We consider 2×2, 3×3, and 4×4 matrices. The idea presented here applies to larger matrices, but the general case will not be discussed in this book. Understanding the presented results and examples should provide a solid foundation for study of the general method…

In Chapter 7 we discussed a decomposition of square matrices that is similar to diagonalization, but can be applied to matrices that are not diagonalizable. In this chapter we consider a decomposition of matrices that are not square matrices that is also similar to diagonalization…

In this chapter we introduce some ideas that are important from the point of view of mathematics as well as applications. The chapter also provides an opportunity to practice applying many of the tools we learned in the first eight chapters of this book.

In this chapter we show that instead of the space ℝⁿ we can consider other sets like the set of all matrices of a given size or the set of all real-valued functions defined on a common domain. It turns out that most results formulated and proven about the space ℝⁿ remain valid in such sets.

In this chapter we discuss some problems that are designed to be solved with the aid of a computer algebra system. We present solutions using Maple, but other systems can be used.

The purpose of this chapter is not only to explain how to facilitate calculations using a computer algebra system, but also to improve understanding of the standard questions and methods of linear algebra. In order to accomplish this goal we present solutions which are more relevant for understanding of the problem, but not necessarily the shortest solutions. We explain every step of the solution and we solve more problems for more difficult topics, like the subspaces of ℝⁿ or Jordan forms. For some problems we give two solutions. For most problems we also verify the result, which emphasizes what has been accomplished by the solution…

The following sections are included:

• Bibliography
• Index