A Portrait of Linear Algebra

Table of Contents
Chapter Zero
Part I: Set Theory and Basic Logic
Part II: Proofs
Summary
Exercises
Chapter One
1.1 The Main Subject: Euclidean Spaces
Summary
Exercises
1.2 The Span of a Set of Vectors
Summary
Exercises
1.3 Euclidean Geometry
Summary
Exercises
1.4 Systems of Linear Equations
Summary
Exercises
1.5 The Gauss-Jordan Algorithm
Summary
Exercises
1.6 Types of Linear Systems
Summary
Exercises
Chapter Two
2.1 Linear Dependence and Independence
Summary
Exercises
2.2 Introduction to Subspaces
Summary
Exercises
2.3 The Fundamental Matrix Spaces
Summary
Exercises
2.4 The Dot Product and Orthogonality
Summary
Exercises
2.5 Orthogonal Complements
Summary
Exercises
2.6 Full-Rank Systems and Dependent Systems
Summary
Exercises
Chapter Three
3.1 Mapping Spaces: Introduction to Linear Transformations
Summary
Exercises
3.2 Rotations, Projections, and Reflections
Summary
Exercises
3.3 Operations on Linear Transformations and Matrices
Summary
Exercises
3.4 Properties of Operations on Linear Transformations and Matrices
Summary
Exercises
3.5 The Kernel and Range; One-to-One and Onto Transformations
Summary
Exercises
3.6 Invertible Operators and Matrices
Summary
Exercises
3.7 Finding the Inverse of a Matrix
Summary
Exercises
3.8 Conditions for Invertibility
Summary
Exercises
Chapter Four
4.1 Axioms for a Vector Space
Summary
Exercises
4.2 Linearity Properties for Finite Sets of Vectors
Summary
Exercises
4.3 A Primer on Infinite Sets
Summary
Exercises
4.4 Linearity Properties for Infinite Sets of Vectors
Summary
Exercises
4.5 Subspaces, Basis and Dimension
Summary
Exercises
4.6 Diagonal, Triangular, and Symmetric Matrices
Summary
Exercises
Chapter Five
5.1 Introduction to General Linear Transformations
Summary
Exercises
5.2 Coordinate Vectors and Matrices for Linear Transformation
Summary
Exercises
5.3 One-to-One and Onto Linear Transformations; Compositions of Linear Transformations
Summary
Exercises
5.4 Isomorphisms
Summary
Exercises
Chapter Six
6.1 The Join and Intersection of Two Subspaces
Summary
Exercises
6.2 Restricting Linear Transformations and the Role of the Rowspace
Summary
Exercises
6.3 The Image and Preimage of Subspaces
Summary
Exercises
6.4 Cosets and Quotient Spaces
Summary
Exercises
6.5 The Three Isomorphism Theorems
Summary
Exercises
Chapter Seven
7.1 Permutations and The Determinant Concept
Summary
Exercises
7.2 A General Determinant Formula
Summary
Exercises
7.3 Properties of Determinants and Cofactor Expansion
Summary
Exercises
7.4 The Adjugate Matrix and Cramer's Rule
Summary
Exercises
7.5 The Wronskian
Summary
Exercises
Chapter Eight
8.1 The Eigentheory of Square Matrices
Summary
Exercises
8.2 The Geometry of Eigentheory and Computational Techniques
Summary
Exercises
8.3 Diagonalization of Square Matrices
Summary
Exercises
8.4 Change of Basis and Linear Transformations on Euclidean Spaces
Summary
Exercises
8.5 Change of Basis for Abstract Spaces and Determinants for Operators
Summary
Exercises
8.6 Similarity and The Eigentheory of Operators
Summary
Exercises
8.7 The Exponential of a Matrix
Summary
Exercises
Chapter Nine
9.1 Axioms for an Inner Product Space
Summary
Exercises
9.2 Geometric Constructions in Inner Product Spaces
Summary
Exercises
9.3 Orthonormal Sets and The Gram-Schmidt Algorithm
Summary
Exercises
9.4 Orthogonal Complements and Decompositions
Summary
Exercises
9.5 Orthonormal Bases and Projection Operators
Summary
Exercises
9.6 Orthogonal Matrices
Key Concepts
Exercises
9.7 Orthogonal Diagonalization of Symmetric Matrices
Summary
Exercises
Chapter Ten
10.1 The Field of Complex Numbers
Summary
Exercises
10.2 Complex Vector Spaces
Summary
Exercises
10.3 Complex Inner Products
Summary
Exercises
10.4 Complex Linear Transformations and The Adjoint
Summary
Exercises
10.5 Normal Matrices
Key Concepts
Exercises
10.6 Schur's Lemma and the Spectral Theorems
Summary
Exercises
10.7 Simultaneous Diagonalization
Summary
Exercises
Glossary of Symbols
Subject Index