Preface

This book is an introduction to linear algebra for pre-calculus students. It is a stand-alone unit in the sense that no prior knowledge of matrices is assumed. Students with experience in general mathematics, up to and including Algebra I, should be able to comprehend the material. However, most students have not had experience with the topics in the latter chapters, so the pace of the course should allow for the students to spend extra time with these chapters. We begin with chapters that explain the matrix operations of addition, subtraction, scalar multiplication, and matrix multiplication. These topics are covered in most pre-calculus texts that are currently in use. This unit also allows the students to explore the notions of inverse, determinant, and consistent and inconsistent systems; these topics are covered in some pre-calculus text books. Our unit also provides the students with an introduction to Markov chains, curve fitting, eigenpairs, and some of the numerical challenges that are encountered when matrices are used to solve real-world problems. These latter topics are rarely addressed in pre-calculus texts. The unit was created from elementary principles with significant input from Rice University faculty and students. Various current texts, recommendations from the National Council of Teachers of Mathematics (NCTM), and the Texas Essential Knowledge and Skills (TEKS) were examined in order to determine which topics should and should not be included in this text.

The state of Texas significantly influences the content of pre-college text books, because a book that is approved for adoption in Texas has a large potential market. In order for a book to be adopted in Texas, all of the TEKS for that course must be covered in the text. Algebra II TEK b2 (A) states that "the student uses tools including matrices ... (to) transform and solve equations." Since these are the only requirements concerning matrices for Texas high-school mathematics books, many books meet these requirements but do not really give the students an adequate understanding of linear algebra. It is true that a college linear algebra text would contain ample detail, but few pre-calculus students have the mathematical maturity necessary to read these texts.

Since most pre-calculus texts only touch on the subject of matrices, one might question the need for a more in-depth study of linear algebra at the pre-calculus level. The NCTM has recognized this need and stated that "matrices and their applications" should receive "increased attention" in high school (Curriculum and Evaluation Standards for School Mathematics, 1989, p. 126). It also could be argued that linear algebra is as important as calculus to many engineers and other scientists. The introduction of linear algebra at the pre-calculus level would give the students a knowledge base on which to build when they study linear algebra in college. The arrays that are studied in linear algebra are of vital importance to computer programmers and computer users. Linear algebra is also central to the computational and mathematical sciences.

In The Psychology of Learning Mathematics, Richard Skemp states "... the learning of mathematics, especially in its early stages and for the average student, [is] very dependent on good teaching" (1987, p. 21). Unfortunately, many teachers have not had much experience, and do not feel entirely comfortable, with linear algebra, so it is difficult for them to teach more than just the procedures of matrix manipulation. Therefore, we have attempted to write this unit so that the students can directly access the material. Since discussions add to, and strengthen, one's understanding of a topic, thought-provoking questions and their answers are provided at the end of each chapter to spark class discussions. To help the teacher, complete solution steps have been provided in addition to the solutions where appropriate.

Skemp also states that "concepts of a higher order than those which people already have cannot be communicated to them by a definition [alone], but only by arranging for them to encounter a suitable collection of examples" (1987, p. 18). For this reason, most new concepts in this unit are presented with an example that builds on the intuition of the student. Then the formal definition is given, and other examples follow to clarify the concept. This helps motivate the students because they can immediately see a use for the concept. It also gives the concept a foundation in the mind of the student. Although the notion of building concepts in this manner seems logical, few text books utilize this approach.

Because many books teach procedures rather than concepts, the students do not receive enough information to expand beyond the examples in the book. For example, some books teach methods which apply only to the special case of 2 by 2 matrices when they address the notions of inverse and determinant. However, this text presents methods for finding inverses and determinants of square matrices of any size. Since the students learn the concepts and these general methods, their knowledge is not restricted by the examples in the book.

Because computers are essential to modern society, computer programming assignments are included at the end of the first three chapters as a means to help the students solidify their knowledge of matrices. In some of the later chapters, students are encouraged to use calculators to help them explore matrices so that they are not tied to problems that can be reasonably computed by hand. The students are not asked to use a particular computer language or a particular calculator, but the code for working programs are provided for the teachers in BASIC and PASCAL.

When new topics are introduced in this unit, they are tied, as much as possible, to previous topics. This is in an attempt to allow the students to appreciate linear algebra as a whole rather than view each chapter as a separate entity. For example, solutions to systems of equations are computed using Gaussian elimination, Gauss-Jordan elimination, and Cramer's rule. The text demonstrates to the students that a form of Gaussian elimination can be used, as an alternative to expansion by minors, to compute determinants. We use these two methods of computing the determinant to discuss efficiency of algorithms so that the students know that finding the correct answer is not the only concern. The students are also told that the determinant of a matrix is the same as the product of its eigenvalues. The relationship between the steady state of a transition matrix and eigenpairs is also demonstrated to the students. These ties and others provide the students with different perspectives from which to view problems.

Most texts do not mention Markov chains, even though they are well within the grasp of pre-calculus students. The NCTM believes that "in grades 9-12, the mathematics curriculum should include topics from discrete mathematics so that all students can represent graphs, matrices, sequences, and recurrence relations" (1989, p. 176). Hopefully, this chapter will catch the attention of the student because a Markov chain is a real application of matrices rather than a contrived book example. This chapter also offers the student a glance into the fascinating world of probability.

Curve fitting is another interesting application of matrix equations, and it can be used immediately in the life of a pre-calculus student. Most pre-calculus students take a laboratory science in which they could use curve fitting to analyze their data. This cross-discipline application also helps the students to view mathematics as a useful tool rather than just a subject to take in school. The NCTM believes that "in grades 9-12, the mathematics curriculum should include the continued study of data analysis and statistics so that all students can use curve fitting to predict from data" (1989, p. 167). Curve fitting is also a bridge between the fields of mathematics and statistics.

Eigenpairs are essentially never found in pre-calculus text books, but they have a wide range of physical applications that could interest students. The computational methods taught in this unit build naturally on previous topics in the text. Because the computation of eigenpairs quickly increases in difficultly as the size of the matrix increases, only simple examples are given in this unit. However, students are introduced to the concept of matrices and to many of their applications, so they will have a foundation on which to build when they study eigenpairs in college.

The chapter entitled "Numerical Challenges" is important to the students' overall knowledge even though the students are not asked to perform computations. This chapter reminds students that the world is not solely comprised of pretty book examples. It also helps dispel the notion that mathematics is only about learning what other people already know. It is good for students to know that many important challenges remain in mathematics and that bright young minds are needed to research these topics.

This entire unit was written so that pre-calculus teachers and students will have a text that clearly and accurately explains the introductory concepts of linear algebra. It explores the topics that are currently addressed in pre-calculus courses, but emphasizes concepts rather than than just procedures. This unit also provides students with many more real-world linear algebra topics to explore than are presented in current texts. It is hoped that this unit will not only help students understand linear algebra, but will also spark an interest in, and an appreciation for, the mathematical sciences.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? - Albert Einstein

It truly is beautiful that the abstract concepts of mathematics can be used to model the world around us. This is astounding because the laws of mathematics were not created with the universe, but have been defined by mankind over the centuries. These laws model the world so well, that people often fail to distinguish between the real situation and the mathematical model that is being used to study it. For example, because matrices can be used to represent a system of equations which model the real world, people often think that the solution to the system will also be the solution to the real-world problem. However, the solution is only as good as the model that was used to represent the problem. Since the world is so complex, mathematical models cannot accurately model every detail of the universe. However, they may come amazingly close and help illuminate many of the mysteries of the universe.