Linear Algebra III

259 Pages·4.164 MB·English

by  Bookboon.com

Linear Algebra III
Advanced topics
Kenneth Kuttler
Download free books at
Kenneth Kuttler
Linear Algebra III Advanced topics
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Linear Algebra III Advanced topics
© 2012 Kenneth Kuttler & Ventus Publishing ApS
ISBN 978-87-403-0242-4
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Linear Algebra III Advanced topics Contents
Contents
Preface Part I
1 Preliminaries Part I
1.1 Sets And Set Notation Part I
1.2 Functions Part I
1.3 The Number Line And Algebra Of The Real Numbers Part I
1.4 Ordered fields Part I
1.5 The Complex Numbers Part I
1.6 Exercises Part I
1.7 Completeness of R Part I
1.8 Well Ordering And Archimedean Property Part I
1.9 Division And Numbers Part I
1.10 Systems Of Equations Part I
1.11 Exercises Part I
1.12 Fn Part I
1.13 Algebra in Fn Part I
1.14 Exercises Part I
1.15 The Inner Product In Fn Part I
1.16 What Is Linear Algebra? Part I
1.17 Exercises Part I
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Linear Algebra III Advanced topics Contents
2 Matrices And Linear Transformations Part I
2.1 Matrices Part I
2.2 Exercises Part I
2.3 Linear Transformations Part I
2.4 Subspaces And Spans Part I
2.5 An Application To Matrices Part I
2.6 Matrices And Calculus Part I
2.7 Exercises Part I
3 Determinants Part I
3.1 Basic Techniques And Properties Part I
3.2 Exercises Part I
3.3 The Mathematical Theory Of Determinants Part I
3.4 The Cayley Hamilton Theorem Part I
3.5 Block Multiplication Of Matrices Part I
3.6 Exercises Part I
4 Row Operations Part I
4.1 Elementary Matrices Part I
4.2 The Rank Of A Matrix Part I
4.3 The Row Reduced Echelon Form Part I
4.4 Rank And Existence Of Solutions To Linear Systems Part I
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Linear Algebra III Advanced topics Contents
4.5 Fredholm Alternative Part I
4.6 Exercises Part I
5 Some Factorizations Part I
5.1 LU Factorization Part I
5.2 Finding An LU Factorization Part I
5.3 Solving Linear Systems Using An LU Factorization Part I
5.4 The PLU Factorization Part I
5.5 Justification For The Multiplier Method Part I
5.6 Existence For The PLU Factorization Part I
5.7 The QR Factorization Part I
5.8 Exercises Part I
6 Linear Programming Part I
6.1 Simple Geometric Considerations Part I
6.2 The Simplex Tableau Part I
6.3 The Simplex Algorithm Part I
6.4 Finding A Basic Feasible Solution Part I
6.5 Duality Part I
6.6 Exercises Part I
7 Spectral Theory Part II
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Linear Algebra III Advanced topics Contents
7.1 Eigenvalues Part II
7.2 Some Applications Of Eigenvalues And Eigenvectors Part II
7.3 Exercises Part II
7.4 Schur’s Theorem Part II
7.5 Trace And Determinant Part II
7.6 Quadratic Forms Part II
7.7 Second Derivative Test Part II
7.8 The Estimation Of Eigenvalues Part II
7.9 Advanced Theorems Part II
7.10 Exercises Part II
8 Vector Spaces And Fields Part II
8.1 Vector Space Axioms Part II
8.2 Subspaces And Bases Part II
8.3 Lots Of Fields Part II
8.4 Exercises Part II
9 Linear Transformations Part II
9.1 Matrix Multiplication As A Linear Transformation Part II
9.2 L(V,W) As A Vector Space Part II
9.3 The Matrix Of A Linear Transformation Part II
9.4 Eigenvalues And Eigenvectors Of Linear Transformations Part II
9.5 Exercises Part II
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Linear Algebra III Advanced topics Contents
10 Linear Transformations Canonical Forms Part II
10.1 A Theorem Of Sylvester, Direct Sums Part II
10.2 Direct Sums, Block Diagonal Matrices Part II
10.3 Cyclic Sets Part II
10.4 Nilpotent Transformations Part II
10.5 The Jordan Canonical Form Part II
10.6 Exercises Part II
10.7 The Rational Canonical Form Part II
10.8 Uniqueness Part II
10.9 Exercises Part II
11 Markov Chains And Migration Processes Part II
11.1 Regular Markov Matrices Part II
11.2 Migration Matrices Part II
11.3 Markov Chains Part II
11.4 Exercises Part II
12 Inner Product Spaces Part II
12.1 General Theory Part II
12.2 The Gram Schmidt Process Part II
12.3 Riesz Representation Theorem Part II
12.4 The Tensor Product Of Two Vectors Part II
12.5 Least Squares Part II
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Linear Algebra III Advanced topics Contents
12.6 Fredholm Alternative Again Part II
12.7 Exercises Part II
12.8 The Determinant And Volume Part II
12.9 Exercises Part II
13 Self Adjoint Operators 11
13.1 Simultaneous Diagonalization 11
13.2 Schur’s Theorem 15
13.3 Spectral Theory Of Self Adjoint Operators 18
13.4 Positive And Negative Linear Transformations 24
13.5 Fractional Powers 27
13.6 Polar Decompositions 30
13.7 An Application To Statistics 35
13.8 The Singular Value Decomposition 35
13.9 Approximation In The Frobenius Norm 38
13.10 Least Squares And Singular Value Decomposition 41
13.11 The Moore Penrose Inverse 42
13.12 Exercises 48
14 Norms For Finite Dimensional Vector Spaces 51
14.1 The p Norms 58
14.2 The Condition Number 62
14.3 The Spectral Radius 65
14.4 Series And Sequences Of Linear Operators 68
14.5 Iterative Methods For Linear Systems 73
14.6 Theory Of Convergence 81
14.7 Exercises 87
15 Numerical Methods For Finding Eigenvalues 97
15.1 The Power Method For Eigenvalues 97
15.2 The QR Algorithm 117
15.3 Exercises 136
Positive Matrices 139
Functions Of Matrices 149
Differential Equations 156
Compactness And Completeness 185
Fundamental Theorem Of Algebra 188
Fields And Field Extensions 192
Bibliography 246
SelectedExercises 249
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Linear Algebra III Advanced topics
To see Chapter 1-6 Download
Linear Algebra I Matrices and Row operations
To see Chapter 7-12 download
Linear Algebra II Spectral Theory and Abstract Vector Spaces
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