LINEAR ALGEBRA A Pathway to Abstract Mathematics 3rd Edition G. Viglino Ramapo College of New Jersey January 2018 Contents CONTENTS CHAPTER 1 MATRICES AND SYSTEMS OF LINEAR EQUATION 1.1 Systems of Linear Equations 1 1.2 Consistent and Inconsistent Systems of Equations 14 Chapter Summary 27 CHAPTER 2 VECTOR SPACES 2.1 Vectors in the plane, and Beyond 31 2.2 Abstract Vector Spaces 40 2.3 Properties of Vector Spaces 51 2.4 Subspaces 59 2.5 Lines and Planes 68 Chapter Summary 75 CHAPTER 3 BASES AND DIMENSION 3.1 Spanning Sets 77 3.2 Linear Independence 86 3.3 Bases 94 Chapter Summary 109 CHAPTER4 LINEARITY 4.1 Linear Transformations 111 4.2 Image and Kernel 124 4.3 Isomorphisms 135 Chapter Summary 149 CHAPTER 5 MATRICES AND LINEAR MAPS 5.1 Matrix Multiplication 151 5.2 Invertible Matrices 165 5.3 Matrix Representation of Linear Maps 178 5.4 Change of Basis 191 Chapter Summary 202 Contents CHAPTER 6 DETERMINANTS AND EIGENVECTORS 6.1 Determinants 205 6.2 Eigenspaces 218 6.3 Diagonalization 233 6.4 Applications 246 6.5 Markov Chains 259 Chapter Summary 275 CHAPTER 7 INNER PRODUCT SPACES 7.1 Dot Product 279 7.2 Inner Product 292 7.3 Orthogonality 301 7.4. The Spectral Theorem 315 Chapter Summary 307 Appendix A Principle of Mathematical Induction Appendix B Solutions to Check Your Understanding Boxes Appendix C Answers to Selected Odd-Numbered Exercises Preface PREFACE There is no mathematical ramp that will enable you to continuously inch your way higher and higher in mathematics. The climb calls for a ladder consisting of discrete steps designed to take you from one mathematical level to another. You are about to take an important step on that lad- der, one that will take you to a plateau where mathematical abstraction abounds. Linear algebra rests on a small number of axioms (accepted rules, or “laws”), upon which a beautiful and practi- cal theory emerges. Technology can be used to reduce the time needed to perform essential but routine tasks. We fea- ture the TI-84+ calculator, but any graphing utility or Computer Algebraic System will do. The real value of whatever technological tool you use is that it will free you to spend more time on the development and comprehension of the theory and its applications. In any event, if you haven’t already discovered in other courses: MATHEMATICS DOES NOT RUN ON BATTERIES Systems of linear equations are introduced and analyzed in Chapter 1. Graphing utilities can be used to solve such systems, but understanding what those solutions represent plays a dominant role throughout the text. We begin Chapter 2 by sowing the seeds for vector spaces in the fertile real number field, where they soon blossom into the concept of an abstract vector. The remainder of Chapter 2 and all of Chapter 3 are dedicated to a study of vector spaces in isolation. Functions from one vector space to another which, in a sense, respect the algebraic structure of those spaces are investigated in Chapters 4 and 5. The sixth chapter focuses on Eigenvalues and Eigenvectors, along with some of their important applications. The first six chapters may provide a full plate for most one-semester courses. If not, then Chap- ter 7 (on inner product spaces) is offered for dessert. We have made every effort to provide a leg-up for the step you are about to take. Our primary goal was to write a readable book, without compromising mathematical integrity. Along the way, you will encounter numerous Check Your Understanding boxes designed to challenge your under- standing of each newly introduced concept. Complete solutions to the problems in those boxes appear in Appendix B, but please don’t be in too much of a hurry to look at our solutions. You should TRY to solve the problems on your own, for it is only through ATTEMPTING to solve a problem that one grows mathematically. In the words of Descartes: We never understand a thing so well, and make it our own, when we learn it from another, as when we have discovered it for ourselves. 1.1 Systems of Linear Equations 1 1 CHAPTER 1 MATRICES AND SYSTEMS OF LINEAR EQUATIONS Much of the development of linear algebra calls for the solution and interpretation of systems of linear equations. While the “solution part” can be relegated to a calculator, the “interpretation part” cannot. We focus on the solution-part of the process in Section 1, and on the more important interpretation-part in Sections 2. §1. SYSTEMS OF LINEAR EQUATIONS To solve the system of equations: 2x +64y – 4z = 2x ++6y 4z = 0 xy++2z = – 2 is to determine values of the variables (or unknowns) x, y, and z for which each of the three equations is satisfied. You certainly solved such systems in earlier courses, and if you take the time to solve the above Brackets are used to denote sets. In particular, system, you will find that it has but one solution: –11– 1 x ===–11 y z –1 . We can also say that the three-tuple denotes the set containing but one element—the ele- –11– 1 is a solution of the given system of equation, and that ment –11– 1 . –11– 1 is its solution set. In general: An (ordered) n-tuple is an expression of the form c1c2 cn , where each ci is a real number (written ci ), for 1 in. We say that the n-tuple c1c2 cn is a solution of the system of m equations in n unknowns: The x s denote variables a11x1 ++a12x2 a1nxn = b1 i (or unknowns), while the a x ++a x a x = b 21 1 22 2 2n n 2 aij ’s and bi ’s are con- . ... stants (or scalars). ... ... am1x1 ++am2x2 amnxn = bm if each equation in the system is satisfied when ci is substi- tuted for xi , for 1 in. The set of all solutions of a system of equations is said to be the solution set of that system. 2 Chapter 1 Matrices and Systems of Linear Equations EQUIVALENT SYSTEMS OF EQUATIONS Consider the system of equations: – 3x +2y = x 1 ---2+ y = --- 2 2 As you know, you can perform certain operations on that system which will not alter its solution set. For example, you can: (1) Interchange the order of the equations: – 3x +2y = x 1 ---2+ y = --- x 1 2 2 ---2+ y = --- 2 2 – 3x +2y = (2) Multiply both sides of the resulting top equation by 2: x 1 ---2+ y = --- x +14y = 2 2 – 3x +2y = – 3x +2y = (3) Multiply the resulting top equation by 3 and add it to the bottom You used this third equation: maneuver a lot when eliminating a variable x +14y = x +14y = from a given system of equations For example: – 3x +2y = 13y = 5 (i) x +13yz– = multiply by 3 3x +312y = (2) 2x –35y +3z = – 3x +2y = (3) – 3x ++y 2z = 2 add: 13y = 5 multiply (1) by -2 and add it to (2) The above three operations, are said to be elementary equation –511y +1z = operations: 10yz–5= multiply (1) by 3 and add it ELEMENTARY OPERATIONS ON to (3) SYSTEMS OF LINEAR EQUATIONS Interchange the order of any two equations in the system. Multiply both sides of an equation by a nonzero number. Add a multiple of one equation to another equation. EQUIVALENT SYSTEM Two systems of equations sharing a common solution set are said to OF EQUATIONS be equivalent. As you may recall: THEOREM 1.1 Performing any sequence of elementary opera- tions on a system of linear equations will result in an equivalent system of equations. 1.1 Systems of Linear Equations 3 AUGMENTED MATRICES Matrices are arrays of numbers (or expressions representing numbers) arranged in rows and columns: 47 17 0 10 2134 10 6 36 5 12 7 4 1 –739 –38 11 2– 12 9 (i) (ii) (iii) (iv) (v) Matrix (i) contains 2 rows and 3 columns and it is said to be a 23 (two-by-three) matrix. Similarly, (ii) is a 32 matrix, and (iii) is a 33 matrix (a square matrix). In general, an mn matrix is a matrix consisting of m rows and n columns. In particular, (iv) is a 13 matrix (a row matrix), and (v) is a 31 matrix (a column matrix). It is often convenient to represent a system of equations in a more com- pact matrix form. The rows of the matrix in Figure 1.1(b), for example, concisely represents the equations in Figure 1.1(a). Note that the vari- ables x, y, and z are suppressed in the matrix form, and that the vertical line recalls the equal sign in the equations. Such a matrix is said to be the augmented matrix associated with the given system of equations. 2x +64y – 4z = 24–6 4 2x ++6y 4z = 0 26 4 0 AUGMENTED MATRIX xy++2z = – 2 11 2– 2 System of Equations Augmented Matrix (a) (b) Figure 1.1 Switching two equations in a system of equations results in the switching of the corresponding rows in the associated augmented matrix. Indeed, each of the three previously introduced elementary equation operations corresponds with one of the following elementary matrix row operations: ELEMENTARY MATRIX ROW OPERATIONS Interchange the order of any two rows in the matrix. Multiply each element in a row of the matrix by a nonzero number. Add a multiple of one row of the matrix to another row of the matrix. The following terminology is motivated by Theorem 1.1: DEFINITION 1.1 Two matrices are equivalent if one can be EQUIVALENT derived from the other by performing elemen- MATRICES tary row operations.