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Linear Algebra Abridged Sheldon Axler This ﬁle is generated from Linear Algebra Done Right (third edition) by excluding all proofs, examples, and exercises, along with most comments. Learning linear algebra without proofs, examples, and exercises is probably impossible. Thus this abridged version should not substitute for the full book. However, this abridged version may be useful to students seeking to review the statements of the main results of linear algebra. As a visual aid, deﬁnitions are in beige boxes and theorems are in blue boxes. The numbering of deﬁnitions and theorems is the same as in the full book. Thus 1.1 is followed in this abridged version by 1.3 (the missing 1.2 corresponds to an example in the full version that is not present here). This ﬁle is available without charge. Users have permission to read this ﬁle freely on electronic devices but do not have permission to print it. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms. A free sample chapter of the full version, and other information, is available at the book’s website: http://linear.axler.net. 13 March 2016 ©2015

Contents 1 Vector Spaces 1 1.A Rn and Cn 2 4 Complex Numbers 2 Lists 4 Fn Digression on Fields 7 1.B Deﬁnition of Vector Space 1.C Subspaces 11 8 Sums of Subspaces 12 Direct Sums 13 2 Finite-Dimensional Vector Spaces 14 2.A Span and Linear Independence 15 Linear Combinations and Span 15 Linear Independence 17 19 2.B Bases 2.C Dimension 21 3 Linear Maps 23 3.A The Vector Space of Linear Maps 24 Deﬁnition and Examples of Linear Maps 24 Algebraic Operations on L.V; W / 24 Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

Contents vii 3.B Null Spaces and Ranges 26 Null Space and Injectivity 26 Range and Surjectivity 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 Invertibility and Isomorphic Vector Spaces 35 35 Invertible Linear Maps Isomorphic Vector Spaces 36 Linear Maps Thought of as Matrix Multiplication 37 Operators 39 3.E Products and Quotients of Vector Spaces 39 Products of Vector Spaces 39 Products and Direct Sums 40 Quotients of Vector 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 Polynomials 49 50 Complex Conjugate and Absolute Value Uniqueness of Coefﬁcients for Polynomials 51 The Division Algorithm for Polynomials 52 Zeros of Polynomials 52 Factorization of Polynomials over C 53 Factorization of Polynomials over R 55 Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

viii Contents 5 Eigenvalues, Eigenvectors, and Invariant Subspaces 57 5.A Invariant Subspaces 58 Eigenvalues and Eigenvectors 59 Restriction and Quotient Operators 60 5.B Eigenvectors and Upper-Triangular Matrices 61 Polynomials Applied to Operators 61 Existence of Eigenvalues Upper-Triangular Matrices 63 62 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 Positive Operators and Isometries 92 Positive Operators 92 Isometries 93 Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

Contents ix 7.D Polar Decomposition and Singular Value Decomposition 94 Polar Decomposition 94 Singular Value Decomposition 95 8 Operators on Complex Vector Spaces 97 8.A Generalized Eigenvectors and Nilpotent Operators 98 Null Spaces of Powers of an Operator 98 Generalized Eigenvectors 99 Nilpotent Operators 100 8.B Decomposition of an Operator 101 Description of Operators on Complex Vector Spaces 101 Multiplicity of an Eigenvalue 102 Block Diagonal Matrices 103 Square Roots 104 8.C Characteristic and Minimal Polynomials 105 The Cayley–Hamilton Theorem 105 The Minimal Polynomial 106 8.D Jordan Form 107 9 Operators on Real Vector Spaces 109 9.A Complexiﬁcation 110 Complexiﬁcation of a Vector Space 110 Complexiﬁcation of an Operator The Minimal Polynomial of the Complexiﬁcation 112 Eigenvalues of the Complexiﬁcation 112 Characteristic Polynomial of the Complexiﬁcation 114 111 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 Trace and Determinant 118 10.A Trace 119 Change of Basis 119 Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

x Contents Trace: 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 Volume 130 Photo Credits 135 Index 136 Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

CHAPTER1 René Descartes explaining his work to Queen Christina of Sweden. Vector spaces are a generalization of the description of a plane using two coordinates, as published by Descartes in 1637. Vector Spaces Linear algebra is the study of linear maps on ﬁnite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will deﬁne vector spaces and discuss their elementary properties. In linear algebra, better theorems and more insight emerge if complex numbers are investigated along with real numbers. Thus we will begin by introducing the complex numbers and their basic properties. We will generalize the examples of a plane and ordinary space to Rn and Cn, which we then will generalize to the notion of a vector space. The elementary properties of a vector space will already seem familiar to you. Then our next topic will be subspaces, which play a role for vector 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). LEARNING OBJECTIVES FOR THIS CHAPTER basic properties of the complex numbers Rn and Cn vector spaces subspaces sums and direct sums of subspaces Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

2 CHAPTER 1 Vector Spaces 1.A Rn and Cn Complex Numbers 1.1 Deﬁnition complex numbers A complex number is an ordered pair .a; b/, where a; b 2 R, but we will write this as a C bi. The set of all complex numbers is denoted by C: C D fa C bi W a; b 2 Rg: Addition and multiplication on C are deﬁned by .a C bi / C .c C d i / D .a C c/ C .b C d /i; .a C bi /.c C d i / D .ac bd / C .ad C bc/iI here a; b; c; d 2 R. If a 2 R, we identify a C 0i with the real number a. Thus we can think of R as a subset of C. We also usually write 0 C bi as just bi, and we usually write 0 C 1i as just i. Using multiplication as deﬁned above, you should verify that i 2 D 1. Do not memorize the formula for the product of two complex numbers; you can always rederive it by recalling that i 2 D 1 and then using the usual rules of arithmetic (as given by 1.3). 1.3 Properties of complex arithmetic commutativity associativity identities ˛ C ˇ D ˇ C ˛ and ˛ˇ D ˇ˛ for all ˛; ˇ 2 C; .˛Cˇ/C D ˛C.ˇC/ and .˛ˇ/ D ˛.ˇ/ for all ˛; ˇ; 2 C; C 0 D and 1 D for all 2 C; for every ˛ 2 C, there exists a unique ˇ 2 C such that ˛ C ˇ D 0; for every ˛ 2 C with ˛ ¤ 0, there exists a unique ˇ 2 C such that ˛ˇ D 1; .˛ C ˇ/ D ˛ C ˇ for all ; ˛; ˇ 2 C. additive inverse multiplicative inverse distributive property Linear Algebra Abridged is generated from Linear Algebra Done Right (by Sheldon Axler, third edition) by excluding all proofs, examples, and exercises, along with most comments. The full version of Linear Algebra Done Right is available at springer.com and amazon.com in both printed and electronic forms.

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