Solution Manual For
Linear Algebra As an Introduction to Abstract Mathematics Isaiah Lankham Bruno
Nachtergaele
Chapter 1-11 With Appendix [A-F]
analytical skills, and subject-specific knowledge. In this essay, we will explore the nature of exams in business, law, and mathematics, their format, and how they evaluate students’ comprehension and application 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
analytical skills, and subject-specific knowledge. In this essay, we will explore the nature of exams in business, law, and mathematics, their format, and how they evaluate students’ comprehension
and application
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]
analytical skills, and subject-specific knowledge. In this essay, we will explore the nature of exams in business, law, and mathematics, their format, and how they evaluate students’ comprehension and application 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
analytical skills, and subject-specific knowledge. In this essay, we will explore the nature of exams in business, law, and mathematics, their format, and how they evaluate students’ comprehension
and application
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
