Solution Manual For
Linear Algebra As an Introduction to Abstract Mathematics Isaiah Lankham Bruno
Nachtergaele
Chapter 1-11 With Appendix [A-F]
.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from basic arithmetic and algebra to more advanced
areas like calculus, statistics, and
Contents
1 What is Linear Algebra? 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 R2 . . . . . . . . . . . . . . . . . . . 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
The Fundamental Theorem of Algebra and Factoring Polynomials 26
3.1 The Fundamental Theorem of Algebra ........................................................................ 26
, 3.2 Factoring polynomials .................................................................................................... 30
Exercises34.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from
basic arithmetic and algebra to more advanced areas like calculus, statistics, and
4 Vector Spaces 36
4.1 Definition of vector spaces ............................................................................................ 36
4.2 Elementary properties of vector spaces ....................................................................... 39
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 ................................................................................................................................ 55
5.4 Dimension........................................................................................................................ 57
Exercises ................................................................................................................................... 61
6 Linear Maps 64
6.1 Definition and elementary properties ........................................................................... 64
6.2 Null spaces...................................................................................................................... 67
6.3 Range ............................................................................................................................... 69
6.4 Homomorphisms ............................................................................................................. 70
6.5 The dimension formula .................................................................................................. 71
6.6 The matrix of a linear map........................................................................................... 73
6.7 Invertibility....................................................................................................................... 78
Exercises ................................................................................................................................... 82
7 Eigenvalues and Eigenvectors 85
7.1 Invariant subspaces ........................................................................................................ 85
7.2 Eigenvalues ...................................................................................................................... 86
7.3 Diagonal matrices........................................................................................................... 89
7.4 Existence of eigenvalues ................................................................................................ 90
7.5 Upper triangular matrices ............................................................................................. 91
7.6 Diagonalization of 2 × 2 matrices and applications .................................................. 96
Exercises ................................................................................................................................... 98
,8 Permutations and the Determinant of a Square Matrix 102
8.1 Permutations ................................................................................................................. 102
8.1.1 Definition of permutations .............................................................................. 102
8.1.2 Composition of permutations ......................................................................... 105
8.1.3 Inversions and the sign of a permutation ...................................................... 107
8.2 Determinants ................................................................................................................. 110
8.2.1 Summations indexed by the set of all permutations .................................... 110
8.2.2 Properties of the determinant ........................................................................ 112
8.2.3 Further properties and applications ............................................................... 115
8.2.4 Computing determinants with cofactor expansions ..................................... 116
Exercises ................................................................................................................................. 118
.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from basic arithmetic and
algebra to more advanced areas like calculus, statistics, and
9 Inner Product Spaces 120
9.1 Inner product ................................................................................................................ 120
9.2 Norms ............................................................................................................................ 122
9.3 Orthogonality................................................................................................................. 124
9.4 Orthonormal bases....................................................................................................... 127
9.5 The Gram-Schmidt orthogonalization procedure ..................................................... 129
9.6 Orthogonal projections and minimization problems ................................................ 132
Exercises ................................................................................................................................. 136
10 Change of Bases 139
10.1 Coordinate vectors ....................................................................................................... 139
10.2 Change of basis transformation................................................................................... 141
Exercises ................................................................................................................................. 145
11 The Spectral Theorem for Normal Linear Maps 147
11.1 Self-adjoint or hermitian operators ............................................................................ 147
11.2 Normal operators ......................................................................................................... 149
11.3 Normal operators and the spectral decomposition ................................................... 151
11.4 Applications of the Spectral Theorem: diagonalization . . . . . . . . . . . . 153
11.5 Positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
11.6 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
11.7 Singular-value decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 159
, Exercises ................................................................................................................................. 161
.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from basic arithmetic and algebra to more advanced
areas like calculus, statistics, and
List of Appendices
A Supplementary Notes on Matrices and Linear Systems 164
A.1 From linear systems to matrix equations ............................................................. 164
A.1.1 Definition of and notation for matrices ......................................................... 165
A.1.2 Using matrices to encode linear systems....................................................... 168
A.2 Matrix arithmetic..................................................................................................... 171
A.2.1 Addition and scalar multiplication ................................................................. 171
A.2.2 Multiplication of matrices ............................................................................... 175
A.2.3 Invertibility of square matrices ....................................................................... 179
A.3 Solving linear systems by factoring the coefficient matrix .................................. 181
A.3.1 Factorizing matrices using Gaussian elimination .......................................... 182
A.3.2 Solving homogeneous linear systems ............................................................. 192
A.3.3 Solving inhomogeneous linear systems .......................................................... 195
A.3.4 Solving linear systems with LU-factorization................................................. 199
A.4 ..................................................................................................... Matrices and linear maps204
A.4.1 The canonical matrix of a linear map ........................................................... 204
A.4.2 Using linear maps to solve linear systems..................................................... 205
A.5 Special operations on matrices ............................................................................... 211
A.5.1 Transpose and conjugate transpose ............................................................... 211
A.5.2 The trace of a square matrix .......................................................................... 212
Exercises ................................................................................................................................. 214
B The Language of Sets and Functions 218
B.1 Sets ................................................................................................................................ 218
B.2 Subset, union, intersection, and Cartesian product .................................................. 220
B.3 Relations ........................................................................................................................ 222
B.4 Functions ....................................................................................................................... 223
C Summary of Algebraic Structures Encountered 226
C.1 Binary operations and scaling operations .................................................................. 226
C.2 Groups, fields, and vector spaces ............................................................................... 229
C.3 Rings and algebras ....................................................................................................... 233
D Some Common Math Symbols and Abbreviations 236
E Summary of Notation Used 243
F Movie Scripts 246
Linear Algebra As an Introduction to Abstract Mathematics Isaiah Lankham Bruno
Nachtergaele
Chapter 1-11 With Appendix [A-F]
.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from basic arithmetic and algebra to more advanced
areas like calculus, statistics, and
Contents
1 What is Linear Algebra? 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 R2 . . . . . . . . . . . . . . . . . . . 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
The Fundamental Theorem of Algebra and Factoring Polynomials 26
3.1 The Fundamental Theorem of Algebra ........................................................................ 26
, 3.2 Factoring polynomials .................................................................................................... 30
Exercises34.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from
basic arithmetic and algebra to more advanced areas like calculus, statistics, and
4 Vector Spaces 36
4.1 Definition of vector spaces ............................................................................................ 36
4.2 Elementary properties of vector spaces ....................................................................... 39
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 ................................................................................................................................ 55
5.4 Dimension........................................................................................................................ 57
Exercises ................................................................................................................................... 61
6 Linear Maps 64
6.1 Definition and elementary properties ........................................................................... 64
6.2 Null spaces...................................................................................................................... 67
6.3 Range ............................................................................................................................... 69
6.4 Homomorphisms ............................................................................................................. 70
6.5 The dimension formula .................................................................................................. 71
6.6 The matrix of a linear map........................................................................................... 73
6.7 Invertibility....................................................................................................................... 78
Exercises ................................................................................................................................... 82
7 Eigenvalues and Eigenvectors 85
7.1 Invariant subspaces ........................................................................................................ 85
7.2 Eigenvalues ...................................................................................................................... 86
7.3 Diagonal matrices........................................................................................................... 89
7.4 Existence of eigenvalues ................................................................................................ 90
7.5 Upper triangular matrices ............................................................................................. 91
7.6 Diagonalization of 2 × 2 matrices and applications .................................................. 96
Exercises ................................................................................................................................... 98
,8 Permutations and the Determinant of a Square Matrix 102
8.1 Permutations ................................................................................................................. 102
8.1.1 Definition of permutations .............................................................................. 102
8.1.2 Composition of permutations ......................................................................... 105
8.1.3 Inversions and the sign of a permutation ...................................................... 107
8.2 Determinants ................................................................................................................. 110
8.2.1 Summations indexed by the set of all permutations .................................... 110
8.2.2 Properties of the determinant ........................................................................ 112
8.2.3 Further properties and applications ............................................................... 115
8.2.4 Computing determinants with cofactor expansions ..................................... 116
Exercises ................................................................................................................................. 118
.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from basic arithmetic and
algebra to more advanced areas like calculus, statistics, and
9 Inner Product Spaces 120
9.1 Inner product ................................................................................................................ 120
9.2 Norms ............................................................................................................................ 122
9.3 Orthogonality................................................................................................................. 124
9.4 Orthonormal bases....................................................................................................... 127
9.5 The Gram-Schmidt orthogonalization procedure ..................................................... 129
9.6 Orthogonal projections and minimization problems ................................................ 132
Exercises ................................................................................................................................. 136
10 Change of Bases 139
10.1 Coordinate vectors ....................................................................................................... 139
10.2 Change of basis transformation................................................................................... 141
Exercises ................................................................................................................................. 145
11 The Spectral Theorem for Normal Linear Maps 147
11.1 Self-adjoint or hermitian operators ............................................................................ 147
11.2 Normal operators ......................................................................................................... 149
11.3 Normal operators and the spectral decomposition ................................................... 151
11.4 Applications of the Spectral Theorem: diagonalization . . . . . . . . . . . . 153
11.5 Positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
11.6 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
11.7 Singular-value decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 159
, Exercises ................................................................................................................................. 161
.________________________________________3. Mathematics Exams3.1. Overview of Mathematics EducationMathematics education spans a wide range of topics, from basic arithmetic and algebra to more advanced
areas like calculus, statistics, and
List of Appendices
A Supplementary Notes on Matrices and Linear Systems 164
A.1 From linear systems to matrix equations ............................................................. 164
A.1.1 Definition of and notation for matrices ......................................................... 165
A.1.2 Using matrices to encode linear systems....................................................... 168
A.2 Matrix arithmetic..................................................................................................... 171
A.2.1 Addition and scalar multiplication ................................................................. 171
A.2.2 Multiplication of matrices ............................................................................... 175
A.2.3 Invertibility of square matrices ....................................................................... 179
A.3 Solving linear systems by factoring the coefficient matrix .................................. 181
A.3.1 Factorizing matrices using Gaussian elimination .......................................... 182
A.3.2 Solving homogeneous linear systems ............................................................. 192
A.3.3 Solving inhomogeneous linear systems .......................................................... 195
A.3.4 Solving linear systems with LU-factorization................................................. 199
A.4 ..................................................................................................... Matrices and linear maps204
A.4.1 The canonical matrix of a linear map ........................................................... 204
A.4.2 Using linear maps to solve linear systems..................................................... 205
A.5 Special operations on matrices ............................................................................... 211
A.5.1 Transpose and conjugate transpose ............................................................... 211
A.5.2 The trace of a square matrix .......................................................................... 212
Exercises ................................................................................................................................. 214
B The Language of Sets and Functions 218
B.1 Sets ................................................................................................................................ 218
B.2 Subset, union, intersection, and Cartesian product .................................................. 220
B.3 Relations ........................................................................................................................ 222
B.4 Functions ....................................................................................................................... 223
C Summary of Algebraic Structures Encountered 226
C.1 Binary operations and scaling operations .................................................................. 226
C.2 Groups, fields, and vector spaces ............................................................................... 229
C.3 Rings and algebras ....................................................................................................... 233
D Some Common Math Symbols and Abbreviations 236
E Summary of Notation Used 243
F Movie Scripts 246
