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 infinitedimensional 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 selfstudy 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 COLORS: cyan magenta yellow black KH final proof: 6-24-16 jjf BOOK: 8.5x11 SPINE: 1.74 for Perfect Binding 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 infinitedimensional 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 selfstudy 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 COLORS: cyan magenta yellow black KH final proof: 6-24-16 jjf BOOK: 8.5x11 SPINE: 1.74 for Perfect Binding 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! To order the print or e-book version of this book, go to: https://he.kendallhunt.com/product/portrait-linear-algebra Cover art: Linear Transformation by Jude Thaddeus Socrates, 2016 www.kendallhunt.com Send all inquiries to: 4050 Westmark Drive Dubuque, IA 52004-1840 Copyright © 2016 by Kendall Hunt Publishing Company ISBN 978-1-4652-9053-3 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner Printed in the United States of America www.ebook3000.com Table of Contents Chapter Zero The Language of Mathematics: Sets, Axioms, Theorems & Proofs Chapter 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 155 Chapter 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 267 Chapter From The Real to The Abstract: General Vector Spaces 3.1 Axioms for a Vector Space 269 270 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 iii 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 391 Chapter 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 445 Chapter 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 501 Chapter 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 575 Chapter 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 iv www.ebook3000.com 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 669 Chapter 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 751 Chapter 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 v www.ebook3000.com 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 vii 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 or 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 , 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 viii www.ebook3000.com Appendix B: Exercises For Exercises 1 to 5: Use the symbols for the logical quantifiers to write the following Axioms of the Real Number System symbolically (see Chapter Zero or Appendix A for their full statements): The Closure Property of Addition The Associative Property of Multiplication The Distributive Property of Multiplication over Addition The Existence of the Multiplicative Identity The Existence of Multiplicative Inverses Prove de Morgan’s Laws using truth tables: For all logical statements p and q: not p and q is logically equivalent to not p or not q and likewise: not p or q is logically equivalent to not p and not q Notice that the truth table for the disjunction contains a lone F entry in the final column, just like the truth table for the implication Use this to rewrite p q in terms of a negation and a disjunction Divisibility: We say that a divides b if b ka for another integer k Rewrite this definition using the universal and existential quantifiers: a , b : a divides b : _ Prime Numbers: We say that p is a prime number if p and the only positive integers n that divide p are and p itself Use the previous Exercise to rewrite this definition using quantifiers 10 Parity: Use the existential quantifier to define the concept of an even integer, and likewise to define an odd integer: a : a is even n : _ a : a is odd 11 Use the previous Exercise to rewrite symbolically the following Theorem, which we saw in Chapter Zero, without using the word odd: For all a, b : If the product a b is odd, then both a and b are odd 12 Use quantifiers to rewrite Goldbach’s Conjecture: Every even integer bigger than can be expressed as the sum of two prime numbers 860 Appendix B: Logical Symbols and Truth Tables www.ebook3000.com Glossary of Symbols Chapter Zero or the empty set or null set “an element of” or member of a set the set of natural numbers the set of integers (from “Zahlen”) the set of rational numbers (from “quotient”) the set of real numbers not p the negation of a logical statement p pq the implication p implies q pq the equivalence p if and only if q p and q the conjunction of p and q p or q the disjunction of p and q “for all”, the universal quantifier “there exists,” the existential quantifier XY X is a subset of Y XY X union Y, or the union of X and Y XY X intersection Y, or the intersection of X and Y XY X minus Y, or the difference between X and Y Chapter n v v , v , , v n n 0, 0, , 0 Euclidean n-space v u v the negative of the vector v the sum of the vectors u and v r v or rv the scalar product of r with v PQ v the vector from a point P to a point Q the length or norm of a vector v Glossary of Symbols an arbitrary vector of n the zero vector of n 861 x 1v x 2v x kv k e , e , , e n i, j and k a linear combination of vectors SpanS the Span of a set of vectors S u v the capital Greek letter “pi,” representing a plane the dot product of the vectors u and v , v du the distance between the vectors u and v v v v n | b the standard unit vectors in n the standard unit vectors in an augmented matrix mn “m by n, ” the dimension of a matrix R i cR i multiply row i by c Ri Rj exchange row i and row j R i R i cR j add c times row j to row i rref the reduced row echelon form of a matrix In Ax the n n identity matrix a matrix product with a column vector v v v n a matrix with vectors assembled in columns bW the subspace symbol W dimW S w a translate (or coset) of a subspace W the orthogonal complement of a subspace W the dimension of a space (or subspace) W with respect to the basis S the coordinate vector of w Chapter T the standard matrix of a linear transformation Z n,m the zero transformation from n into m mn the zero m n matrix I n Sk the identity operator on n the scaling operator S k v kv rot the counterclockwise rotation in by proj x the projection operator onto the x-axis (etc.) refl x the reflection operator across the x-axis (etc.) refl the reflection operator in across 862 Glossary of Symbols www.ebook3000.com T2 T1 the composition of T with T kerT the kernel of a linear transformation T rangeT the range of a linear transformation T nullityT the dimension of kerT rankT the dimension of rangeT T 1 the inverse of an invertible operator T A 1 Diag d , d , , d n A the inverse of an invertible square matrix A a diagonal matrix with diagonal entries d , d , , d n the transpose of a matrix A Chapter a generalized vector addition 0V a generalized scalar multiplication n the space of polynomials of degree at most n FI the space of functions defined on an interval I Mat m, n the space of m n matrices the zero vector of the vector space V the space of positive numbers under multiplication and exponentiation |X | the cardinality of a set X 0 “aleph zero” or “aleph nought,” the cardinality of 1 “aleph one,” the cardinality of CI the space of continuous functions on I C I the space of n-times differentiable functions on I n whose derivatives are all continuous C I the space of infinitely differentiable functions on I IV the identity operator on V Ea v B the evaluation transformation: E a f f a the coordinate matrix of v with respect to the basis B T B,B / the matrix of a transformation T relative to B and B / T B the matrix of an operator T relative to the basis B C B,B / the change of basis matrix from B to B / Glossary of Symbols 863 Chapter VW the join of the subspaces V and W VW the intersection of the subspaces V and W VW the direct sum of the subspaces V and W TV the image of a subspace V of the domain of T 1 T W the preimage of a subspace W of the codomain of T T |U the restriction of T to a subspace U of the domain of T V/W “V modulo W, ” the quotient space of V modulo W Chapter detA or |A| the determinant of the square matrix A i1, i2, , in “sigma,” a permutation of 1 n sgn the sign of the permutation M i,j the i, j-minor of a square matrix A C i,j the i, j-cofactor of a square matrix A Chapter “lambda,” an eigenvalue of a square matrix A p or p A the characteristic polynomial of a square matrix A Eig A, B S the eigenspace of a square matrix A associated to the matrix with columns v i S , where B v , , v n detT the determinant of an operator T AB the square matrix A is similar to B trA the trace of a square matrix A Eig T, the eigenspace of an operator T associated to cof A the cofactor matrix of a square matrix A adj A the adjugate matrix of a square matrix A Chapter u |v the inner product of u and v v the length of v: v d u, v the distance between u and v: d u, v u v W v |v the orthogonal complement of a subspace W proj W the projection operator onto W refl W the reflection operator across W 864 Glossary of Symbols www.ebook3000.com Chapter i the imaginary unit 1 the field of complex numbers z a bi z the complex conjugate of z: z a bi z the norm or length of z: z 0 the complex zero: 0i 1 the complex unit: 0i F an arbitrary field, with zero and unit F and F argz n a2 b2 the argument of z; the angle made by z with the positive real axis the complex vector space of all n tuples of complex numbers n all polynomials of degree at most n with complex coefficients Mat, m, n z |w the space of m n matrices whose entries are from the complex inner product of z and w SpecT the spectrum of an operator T; the set of its eigenvalues , , , k A the Hermitian adjoint of A: A A T the Hermitian adjoint of T: T T Hermn the real vector space of all n n Hermitian matrices over SkewHermn the real vector space of all n n Skew-Hermitian matrices over AB the relation A is unitarily equivalent to B Chapter “Ohm,” the unit of resistance “gamma,” the golden ratio /2 Q x1, , xn a quadratic form in n variables Q the matrix of the quadratic form Q “delta,” the discriminant of a binary quadratic form Qx, y A 0, A 0, etc A is a positive definite (resp semi-definite) matrix, etc Q 0, Q 0, etc Q is a positive definite (resp semi-definite) quadratic form, etc A k the upper left k k submatrix of A i a singular value for A: i U V the singular value decomposition of A A Glossary of Symbols i , where i the pseudoinverse of A: A V U 865 Subject Index Important Axioms, Definitions, Theorems and Algorithms are in bold italics ℵ (aleph) 297 absolutely convergent series 328, 588 Absorption Rule 423 abstract vector space 270 addition of vectors 27, 270 addition of linear transformations 186, 333 additive identity 4, 26, 839 additive inverse 4, 26, 31, 270, 674, 687 additive / additivity property 89, 158, 230, 329, 578, 702, 705 adjoint of a complex matrix 705 adjoint transformation 807 adjugate matrix 488 aleph 297 algebraic multiplicity 530, 707 ambient space 115 angle between vectors 60, 592 antecedent anti-commutativity property 706 argument of a complex number 677 arithmetic of matrices 187 associative properties 4, 31, 199-202, 270, 674, 685, 840 augmented matrix 68 automorphism 376-7 axiom Axiom for a General Line 46 Axiom for a General Plane 48 Axiom for a Plane in Cartesian Space 45 Axiom for Parallel Vectors 30, 277 Axiom of Choice 316 Axioms for the Positive Real Numbers 848 Axioms for the Real Numbers 4, 845, 853 Axioms of a Complex Inner Product Space 696 Axioms of an Abstract Vector Space 270 Axioms of a Vector Space over a Field 685 Axioms of an Inner Product Space back substitution 73 balancing chemical reactions 754 basic box 163 basis / bases of a subspace / vector space 117, 124, 314, 324, 690 basis step in induction 16 Basis Test for Diagonalizability 526 Best Approximation Algorithm 657 bijection / bijective 228, 375, 452, 696 bilinear form 578, 802 binary quadratic form 778 bisymmetric matrix 326 block diagonal form 252, 266, 484, 513, 645, 736 Boolean algebra ℂ 672 cancellation laws / properties 19, 39, 377, 405 Cantor, Georg 307 cardinality of a set 295, 298, 317 Cartesian equation for a plane 46 Cartesian plane 28 Cartesian space 30 case-by-case analysis 13 Cauchy-Schwarz Inequality 59, 60, 591 centralizer of a matrix 327 change of basis matrix 544, 555 characteristic equation / polynomial 505, 568, 703 Chinese Remainder Theorem 254 Cholesky Decomposition 800, 803 circuit analysis 760 closure properties 4, 14, 31, 115, 259, 261, 270, 674, 685 codomain 158, 213, 329, 702 coefficient 33, 90, 284, 300 coefficient matrix 90 cofactor 479 cofactor expansion 480 cofactor matrix 488 866 Subject Index www.ebook3000.com column matrix 68, 159 columnspace 125, 213 Columnspace Test for Consistency 131 commutative properties 4, 31, 55, 199, 207, 270, 674, 685, 840 compatibility requirement for compositions 189 for matrix products 191 complement of two sets Completeness Axiom for Real Numbers 851-52 complex conjugate of a complex number 673, 676 complex conjugate of a complex vector 686 complex Euclidean n-space 685 complex Euclidean inner product 694 complex linear transformation 702 complex numbers 672 complex scalar multiplication 685 complex vector spaces 685 complex zero 672 component of a vector 26 composition of linear transformations 189, 203, 225, 363, 375 conclusion conditionally convergent series 328, 588 conditions in an implication conditions for invertibility 248 conic section 778, 788 conjecture 17 conjugate homogeneity property 706 conjugation by a matrix 563 conjunction 8, 857 consequent consistent system 83, 131 continuous functions 6, 312, 331, 582 contraction operator 164 contrapositive of a logical statement 7, 858 converse of a logical statement 7, 858 coordinate matrix 341 coordinate vector 341 coplanar vectors 93 coset 422 countable sets 297-8 counterexample covariance matrix 803 Subject Index covering 216, 359 Cramer's (Gabriel) Rule 491 cross product 64, 460, 476 cryptography data compression 829 De Moivre's Theorem 678 De Morgan's Laws 9, 398, 623, 860 decode 345 defective matrix 526 degenerate conic 788 dependence (test) equation 90, 287, 688 dependent set of vectors 90 Dependent Sets from Spanning Sets Theorem 105, 316 Descartes' Rule of Signs 518 determinant of a square matrix 448, 461 determinant of an operator 558, 560 diagonal matrix 255, 279, 322, 325, 465 diagonal quadratic form 778, 781 diagonalizable / diagonalize / diagonalization 526, 534, 563, 570, 574, 707 dichotomy property 23, 848 difference of two sets differentiable functions 312 differentiation transformation 331 dilation operator 164 dimension of a matrix 68 dimension of a vector space / subspace 26, 120, 316, 690 Dimension Theorem 130, 146, 153, 214, 248, 361, 616, 703, 810 Direct sum of two or more matrices 252, 266, 484, direct sum of two or more vector spaces 283, 400 directed bipartite graph 453 directed line segment 29 direction vector of a line 46 discriminant of a binary quadratic form 783, 792 disjunction 8, 857 distance between two vectors 61, 590 distributive properties 4, 31, 55, 199, 200 270, 674, 685 divergent series 328 867 domain 158, 329, 702 dot product 54, 144 double adjoint property 705 double conjugate property 610 double implication double index notation 100, 105 double negation property 19 double reciprocal property 20 free variable 69, 87, 90 full rank 126, 133-4, 218 function space 271, 332, 347 Fundamental Matrix Spaces 125 Fundamental Theorem of Algebra 285, 679 Fundamental Theorem of Arithmetic 22 Fundamental Theorem of Linear Algebra 808-9 eigenspace 507 Eigentheory 504, 568, 703 eigenvalues / eigenvectors 504, 568, 703 elementary matrix 163, 182, 238, 466 elementary column operation 247 elementary row operation 70, 238, 466 Elimination Theorem 102, 114, 294 embedding 214, 358 empty set 1, 837 encode 345 Equality of Polynomials 285 Equality of Sets 9, 71 Equality of Spans Theorem 99,100 equality of vectors 26 Equivalence of Cholesky Decomposition 801 equivalence 7, 858 equivalence classes 565 equivalence relation 564, 726 equivalent statements 7, 859 Euclidean Algorithm 254 Euclidean n-space 26 evaluation transformation 330 even permutation 452 Existence of a Basis Theorem 119, 316 Existence of the Empty Set Axiom 3, 837 existential quantifier 3, 856 exponential of a square matrix 540 Extension Theorem 98, 106, 118, 294, 315, 622 Galois Theory 679 Gauss, Carl Friedrich 679 Gauss-Jordan Algorithm / Gaussian Elimination 73, 689 geometric multiplicity 530, 707 Geometric vs Algebraic Multiplicity Theorem 531, 707 given condition Goldbach's conjecture 17 Golden Ratio / Rectangle 771 Gram-Schmidt Algorithm 602, 697 greatest lower bound 851 Group Theory 428 Fibonacci Number 20, 770, 813 Fibonacci Prime Conjecture 20 Field Axioms 4, 674-5, 845 fields 674 finite dimensional vector spaces 316, 690 finite field 683 Fourier series 612 Hermite, Charles 696 Hermitian adjoint 705 Hermitian matrix 712 Hermitian symmetry 696 Hermitian transpose 705 homogeneity property 55, 62, 89, 158, 230, 329, 578, 702 homogeneous system 87, 143 Hooke's Law 827 Hypothesis i imaginary unit 672 idempotent matrix 632 identity matrix 76, 162 identity operator 162, 330, 377 identity property 26 image 158 image of a subspace 412 imaginary axis 677 imaginary part of a complex number 672 imaginary unit i 672 implication 5, 858 improper orthogonal matrix 635 868 Subject Index www.ebook3000.com inconsistent system 83 indefinite matrix 796 Independence of Distinct Eigenspaces 529 independent set of vectors 90, 260, 685 Independent Sets from Spanning Sets Theorem 106, 316 inductive hypothesis / step infinite dimensional vector space 316, 690 infinitude of primes 22 injection / injective linear transformation 216, 358 inner product 578 integer Integer Roots Theorem 518 integer solutions to linear systems 82 integrals as linear transformations 331 integrals as inner products 582 intercepts form of the equation of a plane 53 intersection of two sets 9, 398 intersection of two subspaces 397 Invariance of the Discriminant 783 Invariance of Solution Sets 71 Invariant Properties under Similarity 454 inverse of a composition 250 inverse of a logical statement 7, 858 inverse of a matrix 231, 240 inverse of a matrix product 250 inverse of a reflection operator 234 inverse of an operator 228, 233 inverse property 26 inversion in a permuation 448, 452 invertible linear transformation 228, 375 invertible matrix 231, 448, 477, 490, 560 irrational number isometry 627 isomorphism 228, 375 Isomorphism Theorems 431-441 join of two subspaces 395 kernel of a linear transformation 215, 334, 375, 708 Kernel Test for Injectivity 215, 358 Kirchoff's Current / Voltage Laws 761-2 Subject Index Law of Cosines 56 leading column / leading one / leading variable 69 Least Squares Method / solution / system 653 least upper bound 851 left handed coordinate system 642 left homogeneity property 696 left inverse of a matrix 249 length of a vector 32, 54, 588, 695 Linear Algebra 25 linear combination of linear transformations 193, 333 linear combination of matrices 193 linear combination of vectors 33, 284, 300, 688 linear dependence / independence 90, 260, 685 linear functional 158 linear operator 158, 329 linear transformation 158, 283, 329 line 3, 46 logical operators logical statement / system lower bound 849 lower triangular matrix 260, 322, 325 main diagonal 257 map 158 mathematical research 17 matrices / matrix 68 matrix equation 89 matrix of a basis 544 matrix of a composition 203, 364 matrix of a linear transformation 160, 344, 548, 557 matrix over ℂ 687 matrix product 88, 144, 191, 702 matrix space 274 medians of a triangle 39 Mersenne prime mesh analysis 763 Method of Descent 22 Minimizing Theorem 102, 118, 122, 125, 356, 413, 443, 546 minor 479 869 modus ponens 11 multiplicative identity / inverse 4, 674 multiplicativity property 12, 27, 39, 276 multiplicities 530, 707 Multiplicity Test for Diagonalizability 531 ℕ natural numbers 1, 295, 829 NAND operation 859 necessary condition negation of a logical statement negative definite matrix or quadratic form 796 negative numbers negative semi-definite matrix or quadratic form 796 negative of a matrix 187 negative of a vector 26 nested subspaces 124, 311 Noether, Amalie Emmy 431 non-degeneracy property 55 non-homogeneous system 87 non-trivial solution 87 NOR operation 859 norm of a vector 32, 54, 589 normal line to a plane 58 normal matrix 717 normal system 655, 665 normal vector to a plane 58 normalizing a row 73 not logical operator null set nullity of a matrix / linear transformation 129, 213 nullspace of a matrix 125, 398, 505, 614 octant 30 odd permutation 452 Ohm's Law 760 one sided inverse 249 one-to-one property of linear transformations 214, 358, 362 onto property of linear transformations 216-7, 358, 362 operations on linear transformations 186, 333 Order Axioms for Real Numbers 23, 848 order property of reciprocals 24 ordered n-tuple 26 ordinary differential equation 381 orthogonal basis 601 orthogonal complement 142, 174, 613 orthogonal decomposition 175, 617 orthogonal diagonalization of symmetric matrices 646 orthogonal lines 65 orthogonal lines and planes 65 orthogonal matrix 635 orthogonal operator 639 orthogonal planes 65 orthogonal projection 618 orthogonal set of vectors 599, 609 orthogonal vectors 57, 60, 593 orthonormal basis 601, 616, 625, 627, 639 orthonormal set of vectors 599 overdetermined system 86, 134 ℙⁿ polynomials of degree at most n 271 padding with zeroes 655 parallel lines 48, 52, 65 parallel planes 59, 64, 98 parallel vectors 30, 277 parallelogram law 66 parallelogram principle 29 parameter / parametric equations 42, 45, 47 particular solution of a matrix equation 131 partitioned matrix 88 Pauli matrices 734 permutation matrix 169, 247 permutation of n 451 perpendicular vectors 57, 60, 593 pivot 73 plane 45, 48 polar form of a complex number 677 polynomial curve fitting 383 polynomial evaluation 205, 581 polynomial space 271 polynomial over ℂ 686 positive definite matrix or quadratic form 796 positive rational number 841 positive semi-definite matrix or quadratic form 796 positivity property 55, 578, 695-6 870 Subject Index www.ebook3000.com power of a square matrix or linear transformation 204 powersets 23 predecessor 840, 843 preimage of a subspace 412, 414 Preservation of Subspaces Theorem 412 prime numbers 6, 684 projection matrix 628, 656 projection operator 174-9, 263, 348, 356, 516, 550, 628 proof 11 proof by contradiction 15 proof by contrapositive 14 proof by mathematical induction 15 proof template 35 proper orthogonal matrix 635 proper subspace 115 pseudoinverse 831 pure imaginary complex number 672 pure real complex number 672 Pythagorean Theorem 594 ℚ 2, 295, 306, 675 ℚ⁺ 841 quadratic form 778 quadric surface 778 quantifier QR-Decomposition 663 quotient space 424 ℝ 2, 27, 296 ℝ⁺ 274, 280, 340, 389, 587, 682 ℝ² 28 ℝ³ 30 ℝⁿ 26 range of a linear transformation 213, 334, 375, 702 rank of a linear transformation 213 rank of a matrix 129 rational number 2, 675 Rational Roots Theorem 518 real axis 677 real-conjugate pair 712 real number real part of a complex number 672 Subject Index Really Big Theorem on Invertibility 248, 514 (addenda) reciprocal property 24 recurrence relation 770 reduced row echelon form (rref) 69 reductio ad absurdum 15 reflection operator 164, 172, 174-9, 234, 263, 348, 356, 516, 550, 636, 642 reflexive / reflexivity property 564, 726 relation 564 representative of a coset 422 restriction of a linear transformation 403 right conjugate-homogeneity property 697 right handed coordinate system 642 right inverse of a matrix 249 ring structure 207 roster method for set notation rotation matrix / transformation 170, 512, 636, 644, 788 rotation of a conic section 788 row echelon form 68 row matrix 68 rowspace of a matrix 125, 405 rref (reduced row echelon form) 69, 72, 141 scalar 26 scalar multiplication / product 27, 270, 685 scaling operator 162, 330, 515 Schroeder-Bernstein Theorem 296 Schur, Issai 727 Schur Decomposition / Schur's Lemma 727 shear operator 165, 515, 817, 821 sight-reading the nullspace 130-1 sign of a permutation 452 similarity / similar matrices 563, 566, 574, 725 Simultaneous Diagonalizability Theorem 737, 743, 747 Singular Value Decomposition 817, 819 skew lines 65 Skew-Hermitian matrix 712 skew-symmetric matrix 633, 712 SLn(special linear group) 486 Span of a set of vectors 41, 284, 300, 311, 688 sparse matrix 484 special linear group 486 871 Spectral Theorem for Hermitian / Skew Hermitian / Unitary Matrices 730 Spectral Theorem for Normal Matrices 726 Spectral Theorem for Symmetric Matrices 647, 730 spectrum of an operator 705 square root of a matrix 804 square systems 83, 134 Squeeze Theorem for Real Numbers 24, 850 standard basis vectors 34, 90, 116 standard matrix of a linear transformation 160, 548 standard position 28 subspace 115, 119, 310, 613, 690 substitution principle 12 successor 840, 843 sufficient condition Superposition Principle 763 surjection / surjective 216-7, 358 SVD see Singular Value Decomposition Sylvester's Criterion 798 symmetric equations for a line 43, 47 symmetric matrix 262, 279, 322, 325, 633, 797 symmetry / symmetric property 62, 564, 578, 726 system of linear equations 68 test for pure real / imaginary numbers 676 theorem trace of a matrix 340, 567, 574 transfinite induction (Zorn’s Lemma) 316 transitive / transitivity property 24, 564, 726, 849 translate of a Span / subspace 46, 48, 131, 422 translated vector 28 transpose operation / transpose of a matrix 125, 262 Triangle Inequalities 61, 62, 594 triangular matrices 260, 465, 469, 497, 508 trichotomy property 24 trivial solution 87, 287 trivial subspaces 115 truth tables 856-9 Twin Prime Conjecture 20 Two-for-One Theorem 146, 249, 321 uncountable sets 298, 307 underdetermined system 83, 87, 134, 216 union of two sets 9, 392 Uniqueness of the Additive Identity 38, 275 Uniqueness of the Additive Inverse 19, 38, 276 Uniqueness of the Matrix Inverse 232 Uniqueness of the Multiplicative Inverse 19 Uniqueness of Representation Property of a Basis 34, 124, 314, 324 Uniqueness of the RREF 72, 141 Uniqueness of the Zero Vector 38, 275 unit unit circle / sphere 589 unit vector 32, 54, 589 unitarily diagonalizable matrix 725 unitarily equivalent matrices 725 Unitary Diagonalization Algorithm 731 unitary matrix 715, 725, 730 unitary property 31, 270, 685 universal quantifier 3, 856 upper bound 851 upper triangular matrix 260, 280, 322, 325 Vandermonde determinant 485 vector 26 vector addition / sum 27, 270, 685 vector equation for a line 47 vector space over a field 685 vector subtraction 27 Venn diagram von Neumann, John 837 weight / weighted inner product 579, 587 Wronskian of a set of functions 497 ℤ (Zahlen) 2, 295 zero divisor 207 Zero-Factors Theorem 13, 39, 276 zero matrices 161 zero transformation 161, 330 zero vector / property 26, 55, 270, 685 Zorn's Lemma (transfinite induction) 316 872 Subject Index www.ebook3000.com www.ebook3000.com 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 infinitedimensional 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 selfstudy 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 COLORS: cyan magenta yellow black KH final proof: 6-24-16 jjf BOOK: 8.5x11 SPINE: 1.74 for Perfect Binding ... Binding 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... 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,... so many blessings Jude Thaddeus Socrates Professor of Mathematics Pasadena City College, California June, 2016 vii What Makes This Book Different? A Portrait of Linear Algebra takes a unique approach