A First Course in Linear Algebra

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
	Preface to the Instructor
	Preface to the Student
Acknowledgments
Notation
List of figures
1. Matrices and Vectors
	1.1. Matrices and Linear Systems
	1.2. Row Reduction: Three Elementary Row Operations
	1.3. Vectors in Rn, Linear Combinations and Span
	1.4. Matrix Vector Product and the Equation Ax = b
	1.5. How to Check Your Work
	1.6. Exercises
2. Subspaces in Rn, Basis and Dimension
	2.1. Subspaces in Rn
	2.2. Column Space, Row Space and Null Space of a Matrix
	2.3. Linear Independence
	2.4. Basis
	2.5. Coordinate Systems
	2.6. Exercises
3. Matrix Algebra
	3.1. Matrix Addition and Multiplication
	3.2. Transpose
	3.3. Inverse
	3.4. Elementary Matrices
	3.5. Block Matrices
	3.6. Lower and Upper Triangular Matrices and LU Factorization
	3.7. Exercises
4. Determinants
	4.1. Definition of the Determinant and Properties
	4.2. Alternative Definition and Proofs of Properties
	4.3. Cramer’s Rule
	4.4. Determinants and Volumes
	4.5. Exercises
5. Vector Spaces
	5.1. Definition of a Vector Space
	5.2. Main Examples
	5.3. Linear Independence, Span, and Basis
	5.4. Coordinate Systems
	5.5. Exercises
6. Linear Transformations
	6.1. Definition of a Linear Transformation
	6.2. Range and Kernel of Linear Transformations
	6.3. Matrix Representations of Linear Transformations
	6.4. Change of Basis
	6.5. Exercises
7. Eigenvectors and Eigenvalues
	7.1. Eigenvectors and Eigenvalues
	7.2. Similarity and Diagonalizability
	7.3. Complex Eigenvalues
	7.4. Systems of Differential Equations: the Diagonalizable
	7.5. Exercises
8. Orthogonality
	8.1. Dot Product and the Euclidean Norm
	8.2. Orthogonality and Distance to Subspaces
	8.3. Orthonormal Bases and Gram–Schmidt
	8.4. Isometries, Unitary Matrices and QR Factorization
	8.5. Least Squares Solution and Curve Fitting
	8.6. Real Symmetric and Hermitian Matrices
	8.7. Singular Value Decomposition
	8.8. Exercises
Answers to Selected Exercises
Appendix
	A.1. Some Thoughts on Writing Proofs
		A.1.1. Non-Mathematical Examples
		A.1.2. Mathematical Examples
		A.1.3. Truth Tables
		A.1.4. Quantifiers and Negation of Statements
		A.1.5. Proof by Induction
		A.1.6. Some Final Thoughts
	A.2. Complex Numbers
	A.3. The Field Axioms
Index