A portrait of linear algebra 3rd edition answer key

Linear Algebra

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The Third Edition of A Portrait of Linear Algebra builds on the strengths of the previous editions, providing the student a unified, elegant, modern, and comprehensive introduction:

• emphasizes the reading, understanding, and writing of proofs, and gives students advice on how to master these skills;

• presents a thorough introduction to basic logic, set theory, axioms, theorems, and methods of proof;

• develops the properties of vector and matrix operations as natural extensions of the field axioms for real numbers;

• gives an early introduction of the core concepts of spanning, linear independence, subspaces (including the fundamental matrix spaces and orthogonal complements), basis, dimension, kernel, and range;

• explores linear transformations and their properties by using their correspondence with matrices, fully investigating injective, surjective, and bijective transformations;

• focuses on the derivative as the prime example of a linear transformation on function spaces, establishing the strong connection between the fields of Linear Algebra and Differential Equations;

• comprehensively introduces infinite cardinalities and infinite- dimensional vector spaces;

• thoroughly develops Permutation Theory to completely prove the properties of determinants;

• presents large non-trivial matrices, especially symmetric matrices, that have multi-dimensional eigenspaces;

• rigorously constructs Complex Euclidean Spaces and inner products, with complete proofs of Schur’s Lemma, the Spectral Theorems for normal matrices, and the simultaneous diagonalization of commuting normal matrices;

• proves and applies the Fundamental Theorem of Linear Algebra, and its twin, the Singular Value Decomposition, an essential tool in modern computation;

• presents application topics from Physics, Chemistry, Differential Equations, Geometry, Computer Graphics, Group Theory, Recursive Sequences, and Number Theory;

• includes topics not usually seen in an introductory book, such as the exponential of a matrix, the intersection of two subspaces, the pre-image of a subspace, cosets, quotient spaces, and the Isomorphism Theorems of Emmy Noether, providing enough material for two full semesters;

• features more than 500 additional Exercises since the 2nd Edition, including basic computations, assisted computations, true or false questions, mini-projects, and of course proofs, with multi-step proofs broken down with hints for the student;

• written in a student-friendly style, with precisely stated definitions and theorems, making this book readable for self- study.

The author received his Ph.D. in Mathematics from the California Institute of Technology in 1993, and since then has been a professor at Pasadena City College.

A Portrait of

Linear Algebra Third Edition

Jude Thaddeus Socrates Pasadena City College

Kendall Hunt publishing c o mpany

Jude Thaddeus Socrates and A Portrait of Linear Algebra are on Facebook. Please visit us!

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Table of Contents Chapter Zero. The Language of Mathematics:

Sets, Axioms, Theorems & Proofs 1

Chapter 1. The Canvas of Linear Algebra: Euclidean Spaces and Subspaces 25

1.1 The Main Subject: Euclidean Spaces 26 1.2 The Span of a Set of Vectors 41 1.3 The Dot Product and Orthogonality 54 1.4 Systems of Linear Equations 67 1.5 Linear Systems and Linear Independence 83 1.6 Independent Sets versus Spanning Sets 99 1.7 Subspaces of Euclidean Spaces; Basis and Dimension 115 1.8 The Fundamental Matrix Spaces 125 1.9 Orthogonal Complements 142 A Summary of Chapter 1 155

Chapter 2. Adding Movement and Colors: Linear Transformations on Euclidean Spaces 157

2.1 Mapping Spaces: Introduction to Linear Transformations 158 2.2 Rotations, Projections and Reflections 170 2.3 Operations on Linear Transformations and Matrices 186 2.4 Properties of Operations on Linear Transformations and Matrices 199 2.5 Kernel, Range, One-to-One and Onto Transformations 213 2.6 Invertible Operators and Matrices 228 2.7 Finding the Inverse of a Matrix 238 2.8 Conditions for Invertibility 248 2.9 Diagonal, Triangular, and Symmetric Matrices 256 A Summary of Chapter 2 267

Chapter 3. From The Real to The Abstract: General Vector Spaces 269

3.1 Axioms for a Vector Space 270 3.2 Linearity Properties for Finite Sets of Vectors 284 3.3 Linearity Properties for Infinite Sets of Vectors 295 3.4 Subspaces, Basis and Dimension 310 3.5 Linear Transformations on General Vector Spaces 329

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3.6 Coordinate Vectors and Matrices for Linear Transformations 341 3.7 One-to-One and Onto Linear Transformations;

Compositions of Linear Transformations 358 3.8 Isomorphisms and their Applications 376 A Summary of Chapter 3 391

Chapter 4. Peeling The Onion: The Subspace Structure of Vector Spaces 393

4.1 The Join and Intersection of Two Subspaces 394 4.2 Restricting Linear Transformations and the Role of the Rowspace 403 4.3 The Image and Preimage of Subspaces 412 4.4 Cosets and Quotient Spaces 422 4.5 The Three Isomorphism Theorems 431 A Summary of Chapter 4 445

Chapter 5. From Square to Scalar: Permutation Theory and Determinants 447

5.1 Permutations and The Determinant Concept 448 5.2 A General Determinant Formula 461 5.3 Computational Tools and Properties of Determinants 477 5.4 The Adjugate Matrix and Cramer’s Rule 488 5.5 The Wronskian 497 A Summary of Chapter 5 501

Chapter 6. Painting the Lines: Eigentheory, Diagonalization and Similarity 503

6.1 The Eigentheory of Square Matrices 504 6.2 Computational Techniques for Eigentheory 514 6.3 Diagonalization of Square Matrices 526 6.4 The Exponential of a Matrix 540 6.5 Change of Basis and Linear Transformations on Euclidean Spaces 544 6.6 Change of Basis for Abstract Spaces and Determinants for Operators 555 6.7 Similarity and The Eigentheory of Operators 563 A Summary of Chapter 6 575

Chapter 7. Geometry in the Abstract: Inner Product Spaces 577

7.1 Axioms for an Inner Product Space 578 7.2 Geometric Constructions in Inner Product Spaces 589

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7.3 Orthonormal Sets and The Gram-Schmidt Algorithm 599 7.4 Orthogonal Complements and Decompositions 613 7.5 Orthonormal Bases and Projection Operators 625 7.6 Orthogonal Matrices 635 7.7 Orthogonal Diagonalization of Symmetric Matrices 646 7.8 The Method of Least Squares 653 7.9 The QR-Decomposition 662 A Summary of Chapter 7 669

Chapter 8. Imagine That: Complex Spaces and The Spectral Theorems 671

8.1 The Field of Complex Numbers 672 8.2 Complex Vector Spaces 685 8.3 Complex Inner Products 694 8.4 Complex Linear Transformations and The Adjoint 702 8.5 Normal Matrices 712 8.6 Schur’s Lemma and The Spectral Theorems 725 8.7 Simultaneous Diagonalization 735 A Summary of Chapter 8 751

Chapter 9. The Big Picture: The Fundamental Theorem of Linear Algebra and Applications 753

9.1 Balancing Chemical Equations 754 9.2 Basic Circuit Analysis 760 9.3 Recurrence Relations 770 9.4 Introduction to Quadratic Forms 778 9.5 Rotations of Conics 788 9.6 Positive Definite Quadratic Forms and Matrices 796 9.7 The Fundamental Theorem of Linear Algebra 807 9.8 The Singular Value Decomposition 817 9.9 Applications of the SVD 827

Appendix A: The Real Number System 837 Appendix B: Logical Symbols and Truth Tables 856 Glossary of Symbols 861 Subject Index 866

The Answer Key to the Exercises is available as a free download at: https://he.kendallhunt.com/product/portrait-linear-algebra

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Preface to the 3rd Edition

In the three years since the 2nd Edition of A Portrait of Linear Algebra came out, I have had the privilege of teaching Linear Algebra every semester, and even during most of the summers. All the new ideas, improvements, exercises, and other changes that have been incorporated in the 3rd edition would not have been possible without the lengthy discussions and interactions that I have had with so many wonderful students in these classes, and the colleagues who adopted this book for their own Linear Algebra class. So let me begin by thanking Daniel Gallup, John Sepikas, Lyman Chaffee, Christopher Strinden, Patricia Michel, Asher Shamam, Richard Abdelkerim, Mark Pavitch, David Matthews, and Guoqiang Song, my colleagues at Pasadena City College who have taught out of my book, for sharing their ideas and experiences with me, their encouragement, and suggestions for improving this text. I am certain that if I begin to name all the students who have given me constructive criticisms about the book, I will miss more than just a handful. There have been hundreds of students who have gone through this book, and I learned so much from my conversations with many of them. Often, a casual remark or a simple question would prompt me to rewrite an explanation or come up with an interesting new exercise. Many of these students have continued on to finish their undergraduate careers at four-year institutions, and have begun graduate studies in mathematics or engineering. Some of them have kept in touch with me over the years, and the sweetest words they have said to me is how easily they handled upper-division Linear Algebra classes, thanks to the solid education they received from my book. I give them my deepest gratitude, not just for their thoughts, but also for giving me the best career in the world. It is hard to believe that ten years ago, the idea of this book did not even exist. None of this would have been possible without the help of so many people. Thank you to Christine Bochniak, Beverly Kraus, and Taylor Knuckey of Kendall Hunt for their valuable assistance in bringing the 3rd edition to fruition. Many thanks to my long-suffering husband, my best friend and biggest supporter, Juan Sanchez-Diaz, for patiently accepting all the nights and weekends that were consumed by this book. And thank you to Johannes, for your unconditional love and for making me get up from the computer so we can go for a walk or play with the ball. I would have gone bonkers if it weren’t for you two. To the members of the Socrates and Sanchez families all over the planet, maraming salamat, y muchas gracias, for all your love and support. Thanks to all my colleagues at PCC, my friends on Facebook, and my barkada, for being my unflagging cheering squad and artistic critics. Thanks to my tennis and gym buddies for keeping me motivated and physically healthy. Thank you to my late parents, Dr. Jose Socrates and Dr. Nenita Socrates, for teaching me and all their children the love for learning. And finally, my thanks to our Lord, for showering my life with so many blessings.

Jude Thaddeus Socrates Professor of Mathematics Pasadena City College, California June, 2016

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What Makes This Book Different?

A Portrait of Linear Algebra takes a unique approach in developing and introducing the core concepts of this subject. It begins with a thorough introduction of the field properties for real numbers and uses them to guide the student through simple proof exercises. From here, we introduce the Euclidean spaces and see that many of the field properties for the real numbers naturally extend to the properties of vector arithmetic. The core concepts of linear combinations, spans of sets of vectors, linear independence, subspaces, basis and dimension, are introduced in the first chapter and constantly referenced and reinforced throughout the book. This early introduction enables the student to retain these concepts better and to apply them to deeper ideas. The Four Fundamental Matrix Spaces are encountered at the end of the first Chapter, and transitions naturally into the second Chapter, where we study linear transformations and their standard matrices. The kernel and range of these transformations tells us if our transformations are one-to-one or onto. When they are both, we learn how to find the inverse transformation. We also see that some geometric operations of vectors in 2 or 3 are examples of linear operators. Once these core concepts are firmly established, they can be naturally extended to create abstract vector spaces, the most important examples of which are function spaces, polynomial spaces, and matrix spaces. Linear transformations on finite dimensional vector spaces can again be coded using matrices by finding coordinates for our vectors with respect to a basis. Everything we encountered in the first two chapters can now be naturally generalized. One of the unique features of this book is the use of projections and reflections in 3, with respect to either a line or a plane, in order to motivate some concepts or constructions. We use them to explore the core concepts of the standard matrix of a linear transformation, the matrix of a transformation with respect to a non-standard basis, and the change of basis matrix. In the case of reflection operators, we see them as motivation for the inverse of a matrix, and as an example of an orthogonal matrix. Projection matrices, on the other hand, are good examples of idempotent matrices. The second half of the book goes into the study of determinants, eigentheory, inner product spaces, complex vector spaces, the Spectral Theorems, and the material necessary to understand and prove the Fundamental Theorem of Linear Algebra, and its twin, the Singular Value Decomposition. We also see several applications of Linear Algebra in science, engineering, and other areas of mathematics. Throughout the book, we emphasize clear and precise definitions and proofs of Theorems, constantly encouraging the student to read and understand proofs, and to practice writing their own proofs.

How this Book is Organized

Chapter Zero provides an introduction to sets and set operations, logic, the field axioms for real numbers, and common proof techniques, emphasizing theorems that can be derived from the field axioms. This brief introductory chapter will prepare the student to learn how to read, understand and write basic proofs. We base our development of the main concepts of Linear Algebra on the following definition:

Linear Algebra is the study of vector spaces, their structure, and the linear transformations that map one vector space to another.

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Chapter 1 rigorously examines the archetype vector spaces: Euclidean spaces, their geometry, and the core ideas of spanning, linear independence, subspaces, basis, dimension and orthogonal complements. We will see the Gauss-Jordan Algorithm, the central tool of Linear Algebra, and use it to solve systems of linear equations and investigate the span of a set of vectors. We will also construct the four fundamental matrix spaces: rowspace, columnspace and nullspace for a matrix and its transpose, and find a basis for each space. Chapter 2 introduces linear transformations on Euclidean spaces as encoded by matrices. We will see how each linear transformation determines special subspaces, namely the kernel and the range of the transformation, and use these spaces to investigate the one-to-one and onto properties. We will define basic matrix operations, including a method to find its inverse when this exists. Chapter 3 generalizes the concepts from Chapters 1 and 2 in order to construct abstract vector spaces and linear transformations from one vector space to another. We focus most of our examples on function spaces (in particular, polynomial spaces), and linear transformations connecting them, especially those involving derivatives and evaluations. We will see that in the finite-dimensional case, a linear transformation can be encoded by a matrix as well. By focusing on function spaces preserved by the derivative operator, the strong relationship between Linear Algebra and Differential Equations is firmly established. Chapter 4 investigates the subspace structure of vector spaces, and we will see techniques to fully describe the join and intersection of two subspaces, the image or preimage of a subspace, and the restriction of a linear transformation to a subspace. We will create cosets and quotient spaces, and see one of the fundamental triptychs of modern mathematics: the Isomorphism Theorems of Emy Noether as applied to vector spaces.

Chapter 5 explores the determinant function, its properties, especially its relationship to invertibility, and efficient algorithms to compute it. We will see Cramer’s rule, a technique to solve invertible square systems of equations, albeit not a very practical one. Chapter 6 introduces the eigentheory of operators both on Euclidean spaces as well as abstract vector spaces. We will see when it is possible to encode operators into the simplest possible form, that is, to diagonalize them. We will study the concept of similarity and its consequences. Chapter 7 generalizes geometry on a vector space by imposing an inner product on it. This allows us to introduce the concepts of norm and orthogonality in abstract spaces. We will explore orthonormal bases, the Gram-Schmidt Algorithm, orthogonal matrices, the orthogonal diagonalization of symmetric matrices, the method of least squares, and the QR-decomposition. Chapter 8 applies the constructions thus far to vector spaces over arbitrary fields, especially the field of complex numbers. The main goal of this chapter is to prove the Spectral Theorem of Normal Matrices. One specific case of this Theorem tells us that symmetric matrices can indeed be diagonalized by orthogonal matrices. We also see that commuting diagonalizable matrices can be simultaneously diagonalized by the same invertible matrix, and present an algorithm to do so. Chapter 9 explores some applications of Linear Algebra in science and engineering. We develop the theories of quadratic forms and positive semi-definite matrices. These enable us to prove The Fundamental Theorem of Linear Algebra, an elegant theorem that ties together the four fundamental matrix spaces and the concepts of eigenspaces and orthogonality. Closely connected to this is the Singular Value Decomposition, which has applications in data processing. This book is intended to serve as a text for a standard 15-week semester course in introductory Linear Algebra. However, enough material is included in this text for two full semesters. This book is my vision of what today’s student in science and engineering should know about this elegant field.

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What is New with the Third Edition?

Over 500 new Exercises have been added since the 2nd edition. The last two Sections of Chapter 1 in the 2nd Edition were reorganized into three new sections. Section 1.7 introduces the concept of a subspace of n and proves that every non-zero subspace has a basis, leading us to define the concept of dimension. Section 1.8 introduces the four fundamental matrix spaces and the Dimension Theorem for Matrices, the properties and relevance of these spaces, and how to find a basis for each of them. Section 1.9 focuses on finding a basis for the orthogonal complement of a subspace of n. There are three completely new sections in the 3rd edition: Section 5.5. The Wronskian: a matrix that can determine if a finite set of functions is linearly independent. Section 6.4. The Exponential of a Matrix: a method to compute eA, where A is a diagonalizable square matrix. This computation is particularly important in finding the solutions to a System of Linear Differential Equations. Section 8.7. Simultaneous Diagonalization: an algorithm to find an invertible matrix C that will simultaneously diagonalize two commuting diagonalizable matrices. This is perhaps one of the most elegant ideas presented in this book.

Special Topics and Mini-Projects

Scattered around the Exercises are multi-step problems that guide the student through various topics that probe deeper into Linear Algebra and its connections with Geometry, Calculus, Differential Equations, and other areas of mathematics such as Set Theory, Group Theory and Number Theory.

The Medians of a Triangle: a coordinate-free proof that the three medians of any triangle intersect at a common point which is 2/3 the distance from any vertex to the opposite midpoint (Section 1.1). The Cross Product: used to create a vector orthogonal to two vectors in 3, and proves its other properties using the properties of the 3  3 determinant (Sections 1.3, 5.1 and 5.2). The Uniqueness of the Reduced Row Echelon Form: uses the concepts of the rowspace of a matrix and the Equality of Spans Theorem to prove that the rref of any matrix is unique (Section 1.8). Drawing Three-Dimensional Objects: applies the concept of a projection in order to show how to draw the edges of a 3-dimensional object as perceived from any given direction (Section 2.2). The Center of the Ring of Square Matrices: uses basic matrix products to show that the only n  n matrices that commute with all n  n matrices are the multiples of the identity matrix (Section 2.4). The Kernel and Range of a Composition: proves that the kernel of a composition T2  T1 contains the kernel of T1, and analogously, the range of T2  T1 is contained in the range of T2 (Section 2.5 for Euclidean Spaces and Section 3.7 for arbitrary vector spaces). The Direct Sum of Matrices: explores the properties of matrices in block-diagonal form (Sections 2.8, 2.9, 5.3, 6.1, 7.6, and 8.7). The Chinese Remainder Theorem: introduced and applied to construct invertible 2  2 integer matrices whose inverses also have integer entries (Section 2.8).

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Cantor’s Diagonal Argument: proves that the set of rational numbers is countable by showing how to list its elements in a sequence (Section 3.3). The Countability of Subintervals of the set of Real Numbers: gives a guided proof that all subintervals of  that contains at least two points have the same cardinality as , by explicitly constructing bijections among these subintervals (Section 3.3).

Bisymmetric Matrices: explores the properties and dimensions of this unusual and interesting family of square matrices (Section 3.4). The Centralizer of a Matrix: proves that the set of matrices that commute with a given square matrix forms a vector space, and finds a basis for it in the 2  2 case (Section 3.4). Vector Spaces of Infinite Series: proves that the set of absolutely convergent series form a subspace of the space of all infinite series, whereas conditionally convergent and divergent series are not closed under addition (Section 3.4). We also see a natural inner product which is well-defined on absolutely convergent series but fails for conditionally convergent series (Section 7.1).

Casting Shadows: shows that the shadow on the floor of an image on a window pane is an example of a linear transformation (Section 3.6). The Vandermonde Determinant: applies row and column operations and cofactor expansions to find a closed formula for the Vandermonde Determinant, and applies it to some Wronskian determinants, proving that certain infinite subsets of function spaces are linearly independent (Sections 5.3 and 5.5).

The Special Linear Group of Integer Matrices: introduces the concept of a group, and proves that the set of all n  n matrices with integer entries and determinant 1 form a group under matrix multiplication. This project also proves that SL2 is generated by two special matrices (Section 5.3).

Invertible Triangular Matrices: uses Cramer’s rule to prove that the inverse of an invertible upper triangular matrix is again upper triangular, and analogously for lower triangular matrices (Section 5.4).

Eigenspaces of Matrices Related to Rotation Matrices: although a rotation matrix itself does not have real eigenvalues unless the rotation is by 0 or  radians, performing the reflection across the x-axis followed by a rotation matrix always leads to real eigenvalues, and a basis for the eigenspaces that involve the half-angle formula (Section 6.1).

Properties Preserved by Similarity: proves that similar matrices share attributes such as determinants, invertibility, arithmetic and geometric multiplicities, and diagonalizability. Introduction to Fourier Series: shows that the infinite family of trigonometric functions  sinnx, cosnx |n   are mutually orthogonal under the inner product defined using the integral of their product over 0, 2 (Section 7.3).

De Morgan’s Laws for Subspaces: proves that V W  V W and V W  V W, connecting the ideas of the intersection and join of two subspaces with their orthogonal complements (Section 7.4).

Matrix Decompositions: shows that any square matrix can be decomposed uniquely as the sum of a symmetric and a skew-symmetric matrix, and that the spaces of symmetric and skew-symmetric matrices are orthogonal complements of each other under a naturally defined inner product on all square matrices (Section 7.5).

Right Handed versus Left Handed Orthonormal Bases: uses the cross-product to define and create right-handed orthonormal bases for 3, and relates the concepts of right-handed versus left-handed

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orthonormal bases to proper versus improper orthogonal matrices (Section 7.6).

Rotations in Space: explicitly constructs the matrix of the counterclockwise rotation by an angle  about a fixed unit normal vector n in 3 by elegantly connecting this operator with the concepts of a right-handed coordinate system, orthogonal matrices, and the change of basis formula (Section 7.6).

Finite Fields: introduces finite fields by constructing the addition and multiplication tables for the finite fields /5 and /7 (Section 8.1). The Pauli matrices: an introduction to normal matrices that are important in Quantum Mechanics (Section 8.6).

A Note on Technology

The calculations encountered in modern Linear Algebra would be all but impossible to perform in practice, especially on large matrices, without the advent of the computer. Obviously, it would be tedious to perform calculations on these large matrices by hand. However, we do encourage the student to learn the algorithms and computations first, by practicing on the homework problems by hand (with the help of a scientific calculator, at best), before using technology to perform these computations. It is easy to find free and downloadable software or apps by typing “Linear Algebra Packages” in a search engine. The following computations and algorithms are relevant for this book:  Matrix Arithmetic: Addition, Multiplication, Inverse, Transpose, Determinant;  The Gauss-Jordan Algorithm and the Reduced Row Echelon Form or rref;  Finding a basis for the Rowspace, Columnspace and Nullspace of a Matrix;  Characteristic Polynomials, Eigenvalues and Bases for Eigenspaces;  The QR-decomposition;  The LU-decomposition;  The Singular Value Decomposition (SVD).

Some graphing calculators also provide many of these routines. We leave it to the instructor to decide whether or not these will be allowed or required in the classroom, homework, or examinations.

To the Student

You are about to embark on a journey that will introduce you to the inner workings of mathematics. So far, Calculus has prepared you to be a whiz at computations. Please keep an open mind, though, as you struggle with a very different skill — learning abstractions, theorems and proofs. Read the text several times (preferably before the lecture), and familiarize yourself with key definitions and theorems connecting these definitions and concepts. The Section Summaries and Chapter Summaries should be very useful in this regard. They are not substitutes, though, for reading the entire text, especially the examples and the proofs of theorems, which I encourage you to imitate. When you are asked to prove a theorem in the exercises, identify the key words and the key symbols and write down their precise definitions or meanings. Identify which conditions are given, and what conditions you are trying to prove or show, and then attempt to tie them together into a well-written proof. Be patient with yourself, and don’t give up if you haven’t given it an honest try. I hope you enjoy this experience, and in the end, I hope that you discover the beauty of mathematics.