LinearAbridged pdf

Linear Algebr a Abridged Sheldon Axler This \ufb01le is generated from Linear Algebra Done Right (third edition) by excluding all proofs, examples, and ex ercises, along with most comments. Learning linear algebra without proofs, examples, and e xercises is probably impossible. Thus this abridged v ersion should not substitute for the full book. Ho wev er , this abridged version may be useful to students seeking to re view the statements of the main results of linear algebra. As a visual aid, de\ufb01nitions are in beige boxes and theorems are in blue boxes. The numbering of de\ufb01nitions and theorems is the same as in the full book. Thus 1.1 is followed in this abridged v ersion by 1.3 (the missing 1.2 corresponds to an example in the full v ersion that is not present here). This \ufb01le is av ailable without charge. Users hav e permission to read this \ufb01le freely on electronic de vices but do not ha ve permission to print it. The full version of Linear Alg ebra Done Right is av ailable at springer .com and amazon.com in both printed and electronic forms. A free sample chapter of the full version, and other information, is a vailable at the book\u2019 s website: http://linear .axler .net. 13 March 2016 ©2015 Contents 1 V ector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 4 F n 4 Digression on Fields 7 1.B De\ufb01nition of V ector Space 8 1.C Subspaces 11 Sums of Subspaces 12 Direct Sums 13 2 F inite-Dimensional V ector Spaces 14 2.A Span and Linear Independence 15 Linear Combinations and Span 15 Linear Independence 17 2.B Bases 19 2.C Dimension 21 3 Linear Maps 23 3.A The V ector Space of Linear Maps 24 De\ufb01nition and Examples of Linear Maps 24 Algebraic Operations on L .V ; W / 24 Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic form s. Contents vii 3.B Null Spaces and Ranges 26 Null Space and Injecti vity 26 Range and Surjecti vity 27 Fundamental Theorem of Linear Maps 28 3.C Matrices 29 Representing a Linear Map by a Matrix 29 Addition and Scalar Multiplication of Matrices 30 Matrix Multiplication 32 3.D In vertibility and Isomorphic V ector Spaces 35 In vertible Linear Maps 35 Isomorphic V ector Spaces 36 Linear Maps Thought of as Matrix Multiplication 37 Operators 39 3.E Products and Quotients of V ector Spaces 39 Products of V ector Spaces 39 Products and Direct Sums 40 Quotients of V ector Spaces 41 3.F Duality 44 The Dual Space and the Dual Map 44 The Null Space and Range of the Dual of a Linear Map 45 The Matrix of the Dual of a Linear Map 47 The Rank of a Matrix 48 4 P olynomials 49 Complex Conjugate and Absolute V alue 50 Uniqueness of Coef \ufb01cients for Polynomials 51 The Di vision Algorithm for Polynomials 52 Zeros of Polynomials 52 Factorization of Polynomials o ver C 53 Factorization of Polynomials o ver R 55 Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms. viii Contents 5 Eigen values, Eigen vectors, and In variant Subspaces 57 5.A In variant Subspaces 58 Eigen values and Eigen vectors 59 Restriction and Quotient Operators 60 5.B Eigen vectors and Upper -T riangular Matrices 61 Polynomials Applied to Operators 61 Existence of Eigen values 62 Upper-T riangular Matrices 63 5.C Eigenspaces and Diagonal Matrices 66 6 Inner Product Spaces 69 6.A Inner Products and Norms 70 Inner Products 70 Norms 73 6.B Orthonormal Bases 76 Linear Functionals on Inner Product Spaces 79 6.C Orthogonal Complements and Minimization Problems 80 Orthogonal Complements 80 Minimization Problems 82 7 Operators on Inner Product Spaces 84 7.A Self-Adjoint and Normal Operators 85 Adjoints 85 Self-Adjoint Operators 86 Normal Operators 88 7.B The Spectral Theorem 89 The Complex Spectral Theorem 89 The Real Spectral Theorem 90 7.C Positi ve Operators and Isometries 92 Positi ve Operators 92 Isometries 93 Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms . Contents ix 7.D Polar Decomposition and Singular V alue Decomposition 94 Polar Decomposition 94 Singular V alue Decomposition 95 8 Operators on Complex V ector Spaces 97 8.A Generalized Eigen vectors and Nilpotent Operators 98 Null Spaces of Po wers of an Operator 98 Generalized Eigen vectors 99 Nilpotent Operators 100 8.B Decomposition of an Operator 101 Description of Operators on Complex V ector Spaces 101 Multiplicity of an Eigen value 102 Block Diagonal Matrices 103 Square Roots 104 8.C Characteristic and Minimal Polynomials 105 The Cayley\u2013Hamilton Theorem 105 The Minimal Polynomial 106 8.D Jordan Form 107 9 Operators on Real V ector Spaces 109 9.A Complexi\ufb01cation 110 Complexi\ufb01cation of a V ector Space 110 Complexi\ufb01cation of an Operator 111 The Minimal Polynomial of the Complexi\ufb01cation 112 Eigen values of the Comple xi\ufb01cation 112 Characteristic Polynomial of the Complexi\ufb01cation 114 9.B Operators on Real Inner Product Spaces 115 Normal Operators on Real Inner Product Spaces 115 Isometries on Real Inner Product Spaces 117 10 T race and Determinant 118 10.A T race 119 Change of Basis 119 Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms. x Contents T race: A Connection Between Operators and Matrices 121 10.B Determinant 123 Determinant of an Operator 123 Determinant of a Matrix 125 The Sign of the Determinant 129 V olume 130 Photo Credits 135 Index 136 Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms . C H A P T E R 1 René Descartes explaining his work to Queen Christina of Sweden. V ector spaces are a generalization of the description of a plane using two coor dinates, as published by Descartes in 1637. V ector Spaces Linear algebra is the study of linear maps on \ufb01nite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will de\ufb01ne vector spaces and discuss their elementary properties. In linear algebra, better theorems and more insight emerge if complex numbers are in vestigated along with real numbers. Thus we will begin by introducing the complex numbers and their basic properties. W e will generalize the e xamples of a plane and ordinary space to R n and C n , which we then will generalize to the notion of a vector space. The elementary properties of a vector space will already seem f amiliar to you. Then our next topic will be subspaces, which play a role for v ector spaces analogous to the role played by subsets for sets. Finally , we will look at sums of subspaces (analogous to unions of subsets) and direct sums of subspaces (analogous to unions of disjoint sets). L E A R N I N G O B J E C T I V E S F O R T H I S C H A P T E R basic properties of the complex numbers R n and C n vector spaces subspaces sums and direct sums of subspaces Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms . 2 C H A P T E R 1 V ector Spaces 1.A R n and C n Complex Numbers 1.1 De\ufb01nition complex numbers \ue00f A complex number is an ordered pair .a; b / , where a; b 2 R , but we will write this as a C b i . \ue00f The set of all complex numbers is denoted by C : C D f a C b i W a ; b 2 R g : \ue00f Addition and multiplication on C are de\ufb01ned by .a C b i / C .c C d i / D .a C c / C .b C d /i ; .a C b i /.c C d i / D .ac \ue000 b d / C .a d C b c /i I here a; b ; c ; d 2 R . If a 2 R , we identify a C 0 i with the real number a . Thus we can think of R as a subset of C . W e also usually write 0 C b i as just b i , and we usually write 0 C 1i as just i . Using multiplication as de\ufb01ned abov e, you should verify that i 2 D \ue000 1 . Do not memorize the formula for the product of two comple x numbers; you can always rederi ve it by recalling that i 2 D \ue000 1 and then using the usual rules of arithmetic (as gi ven by 1.3). 1.3 Proper ties of complex arithmetic commutativity \u02db C \u02c7 D \u02c7 C \u02db and \u02db \u02c7 D \u02c7 \u02db for all \u02db; \u02c7 2 C ; associativity .\u02db C \u02c7 / C \ue015 D \u02db C .\u02c7 C \ue015/ and .\u02db \u02c7 /\ue015 D \u02db .\u02c7\ue015/ for all \u02db; \u02c7 ; \ue015 2 C ; identities \ue015 C 0 D \ue015 and \ue015 1 D \ue015 for all \ue015 2 C ; additive in verse for e very \u02db 2 C , there exists a unique \u02c7 2 C such that \u02db C \u02c7 D 0 ; multiplicative in verse for e very \u02db 2 C with \u02db ¤ 0 , there exists a unique \u02c7 2 C such that \u02db \u02c7 D 1 ; distributi ve pr operty \ue015.\u02db C \u02c7 / D \ue015\u02db C \ue015\u02c7 for all \ue015 ; \u02db ; \u02c7 2 C . Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms . S E C T I O N 1.A R n and C n 3 The properties abov e are prov ed using the familiar properties of real numbers and the de\ufb01nitions of complex addition and multiplication. 1.5 De\ufb01nition \ue000 \u02db , subtraction, 1=\u02db , division Let \u02db; \u02c7 2 C . \ue00f Let \ue000 \u02db denote the additi ve in verse of \u02db . Thus \ue000 \u02db is the unique complex number such that \u02db C . \ue000 \u02db / D 0: \ue00f Subtraction on C is de\ufb01ned by \u02c7 \ue000 \u02db D \u02c7 C . \ue000 \u02db /: \ue00f For \u02db ¤ 0 , let 1=\u02db denote the multiplicati ve in verse of \u02db . Thus 1=\u02db is the unique complex number such that \u02db .1=\u02db / D 1: \ue00f Division on C is de\ufb01ned by \u02c7=\u02db D \u02c7 .1=\u02db /: So that we can con veniently make de\ufb01nitions and prov e theorems that apply to both real and complex numbers, we adopt the follo wing notation: 1.6 Notation F Throughout this book, F stands for either R or C . Thus if we prov e a theorem in volving F , we will kno w that it holds when F is replaced with R and when F is replaced with C . Elements of F are called scalars . The word \u201cscalar\u201d, a fancy w ord for \u201cnumber\u201d, is often used when we want to emphasize that an object is a number , as opposed to a vector (v ectors will be de\ufb01ned soon). For \u02db 2 F and m a positi ve inte ger , we de\ufb01ne \u02db m to denote the product of \u02db with itself m times: \u02db m D \u02db \ue001 \ue001 \ue001 \u02db \u201e \u0192\u201a \u2026 m times : Clearly .\u02db m / n D \u02db m n and .\u02db \u02c7 / m D \u02db m \u02c7 m for all \u02db; \u02c7 2 F and all positi ve integers m; n . Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic forms . 4 C H A P T E R 1 V ector Spaces Lists T o generalize R 2 and R 3 to higher dimensions, we \ufb01rst need to discuss the concept of lists. 1.8 De\ufb01nition list, length Suppose n is a nonnegati ve integer . A list of length n is an ordered collection of n elements (which might be numbers, other lists, or more abstract entities) separated by commas and surrounded by parentheses. A list of length n looks like this: .x 1 ; : : : ; x n /: T wo lists are equal if and only if they ha ve the same length and the same elements in the same order . Thus a list of length 2 is an ordered pair , and a list of length 3 is an ordered triple. Sometimes we will use the word list without specifying its length. Re- member , howe ver , that by de\ufb01nition each list has a \ufb01nite length that is a nonnegati ve integer . Thus an object that looks like .x 1 ; x 2 ; : : : /; which might be said to hav e in\ufb01nite length, is not a list. A list of length 0 looks like this: . / . W e consider such an object to be a list so that some of our theorems will not hav e trivial exceptions. Lists dif fer from sets in two ways: in lists, order matters and repetitions hav e meaning; in sets, order and repetitions are irrelev ant. F n T o de\ufb01ne the higher-dimensional analogues of R 2 and R 3 , we will simply replace R with F (which equals R or C ) and replace the 2 or 3 with an arbitrary positi ve inte ger . Speci\ufb01cally , \ufb01x a positiv e integer n for the rest of this section. 1.10 De\ufb01nition F n F n is the set of all lists of length n of elements of F : F n D f .x 1 ; : : : ; x n / W x j 2 F for j D 1; : : : ; n g : For .x 1 ; : : : ; x n / 2 F n and j 2 f 1; : : : ; n g , we say that x j is the j th coordinate of .x 1 ; : : : ; x n / . Linear Algebra Abridg ed is generated from Linear Algebr a Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and e xercises, along with most comments. The full version of Linear Algebra Done Right is av ailable at springer.com and amazon.com in both printed and electronic form s.

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