Linear Algebra : A First Course in Pure and Applied Math

by Goodaire, Edgar G.

• ISBN13: 9780130470171
• ISBN10: 0130470171
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-01-01
• Publisher: Pearson College Div

Summary

This innovative book features an "Active Reading" theme, stressing the learning of proofs by first focusing on reading mathematics. This helps users understand that linear algebra is not just another course in computation. A secondary theme on Least Squares and the "best" solution to Ax = b adds a modern computational flavor that readers will welcome. Key ideas are revisited & reinforced throughout--Linear independence/dependence; eigenvalues/vectors; projection of one vector on another; the plane spanned by vectors.

Table of Contents

Preface	xi
To the Student	xvii
Suggested Lecture Schedule	xix
The Geometry of the Plane and 3-Space	1	(68)
Vectors	1	(15)
Length and Direction	16	(10)
Lines, Planes, and the Cross Product of Vectors	26	(19)
Projections	45	(9)
Euclidean n-Space	54	(15)
Review Exercises	66	(3)
Matrices and Linear Equations	69	(98)
The Algebra of Matrices	69	(19)
The Inverse and Transpose of a Matrix	88	(10)
Systems of Linear Equations	98	(19)
Homogeneous Systems and Linear Independence	117	(8)
The LU Factorization of a Matrix	125	(19)
LDU Factorizations	144	(8)
Finding the Inverse of a Matrix	152	(15)
Review Exercises	163	(4)
Determinants, Eigenvalues, Eigenvectors	167	(50)
The Determinant of a Matrix	167	(10)
Properties of Determinants	177	(16)
The Eigenvalues and Eigenvectors of a Matrix	193	(10)
Similarity and Diagonalization	203	(14)
Review Exercises	214	(3)
Vector Spaces	217	(68)
The Theory of Linear Equations	217	(16)
Basic Terminology and Concepts	233	(20)
Basis and Dimension; Rank and Nullity	253	(21)
One-Sided Inverses	274	(11)
Review Exercises	282	(3)
Linear Transformations	285	(50)
Linear Transformations of Rn	285	(16)
Matrix Multiplication Revisited	301	(7)
The Matrices of a Linear Transformation	308	(13)
Changing Coordinates	321	(14)
Review Exercises	331	(4)
Orthogonality	335	(66)
Projection Matrices and Least Squares Approximation	335	(18)
The Gram-Schmidt Algorithm and QR Factorization	353	(18)
Orthogonal Subspaces and Complements	371	(16)
The Pseudoinverse of a Matrix	387	(14)
Review Exercises	398	(3)
The Spectral Theorem	401	(56)
Complex Numbers and Vectors	401	(14)
Complex Matrices	415	(8)
Unitary Diagonalization	423	(18)
Real Symmetric Matrices	441	(7)
The Singular Value Decomposition	448	(9)
Review Exercises	455	(2)
Applications	457
Data Fitting	457	(5)
Linear Recurrence Relations	462	(6)
Markov Processes	468	(6)
Quadratic Forms and Conic Sections	474	(5)
Graphs	479
Appendix: Show and Prove	1	(1)
Solutions to True/False Questions and Selected Exercises	1	(1)
Glossary	1	(1)
Index	1

Excerpts

Why yet another book in linear algebra? At many universities, calculus, the usual prerequisite for linear algebra, has become a "course without proofs," with minimal reading requirements. In fact, students who are "successful" in calculus have developed the wonderful ability to match examples to homework with almost no reading. So, if one wishes to keep the algebra (vector spaces, linear transformations) inlinear algebra,students need to be taught not only how to handle proofs, but more fundamentally they need to be taught how to read mathematics. My goal in this book is to use an intrinsically interesting subject to show students that there is more to mathematics than numbers. The presentation in this text is frequently interrupted by "Reading Challenges," little questions a student can use to see if she really does understand the point just made. These Reading Challenges promote active reading. The goal is to get students reading actively by questioning the material as they read. Every section in this book concludes with answers to these Reading Challenges followed by a number of "True/False" questions that test definitions and concepts, not dexterity with numbers or buttons. Every exercise set concludes with "Critical Reading" exercises, which include harder problems, yes, but also problems that are easily solved if understood. Before the review exercises at the end of a chapter, there is a list of key words and concepts that have arisen in the chapter that I hope students will use as a mental check that they have understood the chapter. There are several reasons for the "pure & applied" subtitle. While a pure mathematician myself, 1 have lost sympathy with the dry abstract vector space/linear transformation approach of many books that requires maturity which the typical sophomore does not have. My approach to linear algebra is through matrices. Essentially all vector spaces are (subspaces of) Euclideann-space. Other standard examples, like polynomials and matrices, are merely mentioned. In my opinion, when students are thoroughly comfortable withR n , they have little trouble transferring concepts to general vector spaces. The emphasis on matrices and matrix factorizations, the recurring theme of projections starting in Chapter 1, and the idea of the "best (least squares) solution" to Ax=baround which Chapter 6 revolves, make this book more applied than most of its competition. However, it does not take most readers long to discover the rigor and maturity that is required to move through the pages of this text. Whether a mathematical argument is always labeled "proof," virtually no statements are made without justification and a large number of exercises are of the "show or prove" variety. Since students tend to find such exercises difficult, I have included an appendix entitled "Show and Prove" designed to teach students how to write a mathematical proof and I tempt them actually to read this section by promising that they will find there the solutions to a number of exercises in the text itself! So despite its applied flavor, I believe that a linear algebra course that uses this book would serve well as the "introduction to proof' transition course that is now a part of many degree programs. Organization, Philosophy, Style It is somewhat atypical to begin with, even to include in a linear algebra course, a chapter on vector geometry of the plane and 3-space. My purpose is twofold. Such an introduction allows the immediate introduction of the terminology of linear algebra--linear combination, span (the noun and the verb), linear dependence and independence--in concrete settings that the student can readily understand. Furthermore, the student is quickly alerted to the fact that this course is not about manipulating numbers, that serious thinking and critical reading will be required to succeed. Many exercise sets cont