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