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# mat67_course_notes_2up - Linear Algebra As an Introduction...

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Linear Algebra As an Introduction to Abstract Mathematics Lecture Notes for MAT67 University of California, Davis written Fall 2007, last updated November 19, 2009 Isaiah Lankham Bruno Nachtergaele Anne Schilling Copyright c circlecopyrt 2007 by the authors. These lecture notes may be reproduced in their entirety for non-commercial purposes. Contents 1 What is Linear Algebra? 1 1.1 Introduction to MAT 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 What is Linear Algebra? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Non-linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.3 Linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.4 Applications of linear equations . . . . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Introduction to Complex Numbers 11 2.1 Definition of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations on complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Addition and subtraction of complex numbers . . . . . . . . . . . . . 12 2.2.2 Multiplication and division of complex numbers . . . . . . . . . . . . 13 2.2.3 Complex conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.4 The modulus (a.k.a. norm, length, or magnitude) . . . . . . . . . . . 16 2.2.5 Complex numbers as vectors in R 2 . . . . . . . . . . . . . . . . . . . 18 2.3 Polar form and geometric interpretation for C . . . . . . . . . . . . . . . . . 19 2.3.1 Polar form for complex numbers . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Geometric multiplication for complex numbers . . . . . . . . . . . . . 20 2.3.3 Exponentiation and root extraction . . . . . . . . . . . . . . . . . . . 21 2.3.4 Some complex elementary functions . . . . . . . . . . . . . . . . . . . 22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ii

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3 The Fundamental Theorem of Algebra and Factoring Polynomials 26 3.1 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . 26 3.2 Factoring polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Vector Spaces 36 4.1 Definition of vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Elementary properties of vector spaces . . . . . . . . . . . . . . . . . . . . . 38 4.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Sums and direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Span and Bases 48 5.1 Linear span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Linear Maps 62 6.1 Definition and elementary properties . . . . . . . . . . . . . . . . . . . . . . 62 6.2 Null spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.5 The dimension formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.6 The matrix of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.7 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 Eigenvalues and Eigenvectors 82 7.1 Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.2 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.3 Diagonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.4 Existence of eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 iii 7.5 Upper triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.6 Diagonalization of 2 × 2 matrices and Applications . . . . . . . . . . . . . . 93 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8 Permutations and the Determinant of a Square Matrix 99 8.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.1.1 Definition of permutations . . . . . . . . . . . . . . . . . . . . . . . . 99 8.1.2 Composition of permutations . . . . . . . . . . . . . . . . . . . . . . 103 8.1.3 Inversions and the sign of a permutation . . . . . . . . . . . . . . . . 105 8.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2.1 Summations indexed by the set of all permutations . . . . . . . . . . 107 8.2.2 Properties of the determinant . . . . . . . . . . . . . . . . . . . . . . 109 8.2.3 Further properties and applications . . . . . . . . . . . . . . . . . . . 112 8.2.4 Computing determinants with cofactor expansions . . . . . . . . . . . 113 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9 Inner Product Spaces 117 9.1 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.4 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 9.5 The Gram-Schmidt orthogonalization procedure . . . . . . . . . . . . . . . . 126 9.6 Orthogonal projections and minimization problems . . . . . . . . . . . . . . 128 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10 Change of Bases 136 10.1 Coordinate vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10.2 Change of basis transformation . . . . . . . . . . . . . . . . . . . . . . . . . 138 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 11 The Spectral Theorem for Normal Linear Maps 144 11.1 Self-adjoint or hermitian operators . . . . . . . . . . . . . . . . . . . . . . . 144 11.2 Normal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11.3 Normal operators and the spectral decomposition . . . . . . . . . . . . . . . 148 iv
11.4 Applications of the Spectral Theorem: diagonalization . . . . . . . . . . . . 150 11.5 Positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 11.6 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 11.7 Singular-value decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Supplementary Notes on Matrices and Linear Systems 161 12.1 From linear systems to matrix equations

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