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## Über dieses Buch

In its second edition, this textbook offers a fresh approach to matrix and linear algebra. Its blend of theory, computational exercises, and analytical writing projects is designed to highlight the interplay between these aspects of an application. This approach places special emphasis on linear algebra as an experimental science that provides tools for solving concrete problems.

The second edition’s revised text discusses applications of linear algebra like graph theory and network modeling methods used in Google’s PageRank algorithm. Other new materials include modeling examples of diffusive processes, linear programming, image processing, digital signal processing, and Fourier analysis. These topics are woven into the core material of Gaussian elimination and other matrix operations; eigenvalues, eigenvectors, and discrete dynamical systems; and the geometrical aspects of vector spaces.

Intended for a one-semester undergraduate course without a strict calculus prerequisite, Applied Linear Algebra and Matrix Analysis augments the key elements of linear algebra with a wide choice of optional sections. With the book’s selection of applications and platform-independent assignments, instructors can tailor the curriculum to suit specific interests and ensure students across various disciplines are equipped with the powerful tools of linear algebra.

## Inhaltsverzeichnis

### Chapter 1. LINEAR SYSTEMS OF EQUATIONS

Welcome to the world of linear algebra. The two central problems about which much of the theory of linear algebra revolves are the problem of finding all solutions to a linear system and that of finding an eigensystem for a square matrix. The latter problem will not be encountered until Chapter 5; it requires some background development and the motivation for this problem is fairly sophisticated. By contrast, the former problem is easy to understand and motivate. As a matter of fact, simple cases of this problem are a part of most high-school algebra backgrounds. We will address the problem of existence of solutions for a linear system and how to solve such a system for all of its solutions. Examples of linear systems appear in nearly every scientific discipline; we touch on a few in this chapter.
Thomas S. Shores

### Chapter 2. MATRIX ALGEBRA

In Chapter 1 we used matrices and vectors as simple storage devices. In this chapter matrices and vectors take on a life of their own. We develop the arithmetic of matrices and vectors. Much of what we do is motivated by a desire to extend the ideas of ordinary arithmetic to matrices.
Thomas S. Shores

### Chapter 3. VECTOR SPACES

It is hard to overstate the importance of the idea of a vector space, a concept that has found application in mathematics, engineering, physics, chemistry, biology, the social sciences, and other areas. What we encounter is an abstraction of the idea of vector space that we studied in calculus or high school geometry. These “geometrical vectors” can easily be visualized. In this chapter, abstraction will come in two waves. The first wave, which could properly be called generalization, consists in generalizing the familiar ideas of geometrical vectors of calculus to vectors of size greater than three.
Thomas S. Shores

### Chapter 4. GEOMETRICAL ASPECTS OF STANDARD SPACES

The standard vector spaces have many important extra features that we have largely ignored up to this point. These extra features made it possible to do sophisticated calculations in the spaces and enhance our insight into vector spaces by appealing to geometry. For example, in the geometrical spaces $$\mathbb {R}^{2}$$ and $$\mathbb {R}^{3}$$ that were studied in algebra and calculus, it was possible to compute the length of a vector and angles between vectors. These are visual concepts that feel very comfortable to us. In this chapter we generalize these ideas to standard vector spaces and their subspaces. We will abstract these ideas to general vector spaces in Chapter 6.
Thomas S. Shores

### Chapter 5. THE EIGENVALUE PROBLEM

The first major problem of linear algebra is to understand how to solve the basis linear system $$A\mathbf {x}=\mathbf {b}$$ and what the solution means. We have explored this system from three points of view: In Chapter 1 we approached the problem from an operational point of view and learned the mechanics of computing solutions. In Chapter 2, we took a more sophisticated look at the system from the perspective of matrix theory. Finally, in Chapter 3, we viewed the problem from the vantage of vector space theory.
Thomas S. Shores

### Chapter 6. GEOMETRICAL ASPECTS OF ABSTRACT SPACES

Two basic ideas that we learn in geometry are those of length of a line segment and angle between lines. We have already seen how to extend these ideas to the standard vector spaces. The objective of this chapter is to extend these powerful ideas to general linear spaces. A surprising number of concepts and techniques that we learned in a standard setting can be carried over, almost word for word, to more general vector spaces. Once this is accomplished, we will be able to use our geometrical intuition in entirely new ways. For example, we will be able to have notions of size (length) and perpendicularity for nonstandard vectors such as functions in a function space. We will be able to give a sensible meaning to the size of the error incurred in solving a linear system with finite-precision arithmetic. We shall see that there are many more applications of this abstraction.
Thomas S. Shores

### Backmatter

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