Kuttler K. Linear Algebra And Analysis

New York: Kuttler, 2018. — 619 p.Some Prerequisite Topics
Sets And Set Notation
The Schroder Bernstein Theorem
Equivalence Relations
Well Ordering And Induction
The Complex Numbers And Fields
Polar Form Of Complex Numbers
Roots Of Complex Numbers
The Quadratic Formula
The Complex Exponential
The Fundamental Theorem Of Algebra
Ordered Fields
Polynomials
Examples Of Finite Fields
Division Of Numbers
The Field Zp
Some Topics From Analysis
lim sup and lim inf
ExercisesLinear Algebra For Its Own Sake
Systems Of Linear Equations
Elementary Operations
Gauss Elimination
Exercises
Vector Spaces
Linear Combinations Of Vectors, Independence
Subspaces
Exercises
Polynomials And Fields
The Algebraic Numbers
The Lindemannn Weierstrass TheoreExercises
Linear Transformations
L(V, W) As A Vector Space
The Matrix Of A Linear Transformation
Rotations About A Given Vector∗Exercises
Direct Sums And Block Diagonal Matrices
A Theorem Of Sylvester, Direct Sums
Finding The Minimum Polynomial
Eigenvalues And Eigenvectors Of Linear Transformations
A Formal Derivative, Diagonalizability
Exercises
Canonical Forms
Cyclic Sets
The Rational Canonical Form
Nilpotent Transformations
The Jordan Canonical Form
Exercises
Companion Matrices
Exercises
Determinants
The Function sgn
The Definition Of The Determinant
A Symmetric Definition
Basic Properties Of The Determinant
Expansion Using Cofactors
A Formula For The Inverse
Rank Of A Matrix
Summary Of Determinants
The Cayley Hamilton Theorem
Exercises
Modules And Rings ∗
Integral Domains And The Ring Of Polynomials
Modules And Decomposition Into Cyclic Sub-Modules
A Direct Sum Decomposition
Quotients
Cyclic Decomposition
Uniqueness
Canonical Forms
Exercises
Related Topics
The Symmetric Polynomial Theorem
Transcendental Numbers
The Fundamental Theorem Of Algebra
More On Algebraic Field ExtensionsAnalysis And Geometry In Linear Algebra
Normed Linear Spaces
Metric Spaces
Open And Closed Sets, Sequences, Limit Points, Completeness
Cauchy Sequences, Completeness
Closure Of A Set
Continuous Functions
Separable Metric Spaces
Compact Sets
Lipschitz Continuity And Contraction Maps
Convergence Of Functions
Connected Sets
Subspaces Spans And Bases
Inner Product And Normed Linear Spaces
The Inner Product In Fn
General Inner Product Spaces
Normed Vector Spaces
The p Norms
Orthonormal Bases
Equivalence Of Norms
Norms On L(X, Y )
The Heine Borel Theorem
Limits Of A Function
Exercises
Limits Of Vectors And Matrices
Regular Markov Matrices
Migration Matrices
Absorbing States
Positive Matrices
Functions Of Matrices
Exercises
Inner Product Spaces
Orthogonal Projections
Riesz Representation Theorem, Adjoint Map
Least Squares
Fredholm Alternative
The Determinant And Volume
Exercises
Matrices And The Inner Product
Schur’s Theorem, Hermitian Matrices
Quadratic Forms
The Estimation Of Eigenvalues
Advanced Theorems
Exercises
The Right Polar Factorization∗An Application To Statistics
Simultaneous Diagonalization
Fractional Powers
Spectral Theory Of Self Adjoint Operators
Positive And Negative Linear Transformations
The Singular Value DecompositionApproximation In The Frobenius Norm
Least Squares And Singular Value Decomposition
The Moore Penrose Inverse
The Spectral Norm And The Operator Norm
The Positive Part Of A Hermitian Matrix
Exercises
Analysis Of Linear Transformations
The Condition Number
The Spectral Radius
Series And Sequences Of Linear Operators
Iterative Methods For Linear Systems
Theory Of Convergence
Exercises
Numerical Methods, Eigenvalues
The Power Method For Eigenvalues
The Shifted Inverse Power Method
The Explicit Description Of The Method
Automation With MatLAB
Complex Eigenvalues
Rayleigh Quotients And Estimates for Eigenvalues
The QR Algorithm
Basic Properties And Definition
The Case Of Real Eigenvalues
The QR Algorithm In The General Case
Upper Hessenberg Matrices
ExercisesAnalysis Which Involves Linear Algebra
The Derivative, A Linear Transformation
Basic Definitions
The Chain Rule
The Matrix Of The Derivative
A Mean Value Inequality
Existence Of The Derivative, C
FunctiBrouwer Fixed Point Theorem Rn
Simplices And Triangulations
Labeling Vertices
The Brouwer Fixed Point Theorem
Invariance Of Domain
Tensor Products
The Norm In Tensor Product Space
The Taylor Formula And Tensors
Exercises
Abstract Measures And Measurable Functions
Simple Functions And Measurable Functions
Measures And Their Properties
Dynkin’s Lemma
Measures And Regularity
When Is A Measure A Borel Measure?
Measures And Outer Measures
Exercises
An Outer Measure On P (R)
Measures From Outer Measures
One Dimensional Lebesgue Stieltjes Measure
Exercises
The Abstract Lebesgue Integral
Definition For Nonnegative Measurable Functions
Riemann Integrals For Decreasing Functions
The Lebesgue Integral For Nonnegative Functions
The Lebesgue Integral For Nonnegative Simple Functions
The Monotone Convergence Theorem
Other Definitions
Fatou’s Lemma
The Integral’s Righteous Algebraic Desires
The Lebesgue Integral, LThe Dominated Convergence Theorem
Exercises
Measures From Positive Linear Functionals
Lebesgue Measure On Rn,Fubini’s Theorem
The Besicovitch Covering Theorem
Change Of Variables, Linear Map
Vitali Coverings
Change Of Variables
Exercises
The Lp Spaces
Basic Inequalities And Properties
Density Considerations
Separability
Continuity Of Translation
Mollifiers And Density Of Smooth Functions
Fundamental Theorem Of Calculus For Radon Measures
ExercisesRepresentation Theorems
Basic Theory
Radon Nikodym Theorem
Improved Change Of Variables Formula
Vector Measures
Representation Theorems For The Dual Space Of LpThe Dual Space Of C (X)
ExercisesAppendix
The Cross Product
The Box Product
The Distributive Law For Cross Product
Weierstrass Approximation Theorem
Functions Of Many Variables
Tietze Extension Theorem
The Hausdorff Maximal Theorem
The Hamel Basis
Exercises