Introduction to Linear and Matrix Algebra

Preface
	The Purpose of this Book
	Features of this Book
	Focus
	Notes in the Margin
	Exercises
	To the Instructor and Independent Reader
	Sectioning
	Extra Topic Sections
	Lead-in to Advanced Linear and Matrix Algebra
	Acknowledgments
Preface The Purpose of this Book Features of this Book To the Instructor and Independent ReaderChapter 1: Vectors and Geometry1.1 Vectors and Vector Operations1.1.1 Vector Addition1.1.2 Scalar Multiplication1.1.3 Linear Combinationsƒƒƒ‚Exercises1.2 Lengths, Angles, and the Dot Product1.2.1 The Dot Product1.2.2 Vector Length1.2.3 The Angle Between Vectorsƒƒƒ‚Exercises1.3 Matrices and Matrix Operations1.3.1 Matrix Addition and Scalar Multiplication1.3.2 Matrix Multiplication1.3.3 The Transpose1.3.4 Block Matricesƒƒƒ‚Exercises1.4 Linear Transformations1.4.1 Linear Transformations as Matrices1.4.2 A Catalog of Linear Transformations1.4.3 Composition of Linear Transformationsƒƒƒ‚Exercises1.5 Summary and Review1.A Extra Topic: Areas, Volumes, and the Cross Product1.B Extra Topic: Paths in GraphsChapter 2: Linear Systems and Subspaces2.1 Systems of Linear Equations2.1.1 Matrix Equations2.1.2 Row Echelon Form2.1.3 Gaussian Elimination2.1.4 Solving Linear Systems2.1.5 Applications of Linear Systemsƒƒƒ‚Exercises2.2 Elementary Matrices and Matrix Inverses2.2.1 Elementary Matrices2.2.2 The Inverse of a Matrix2.2.3 A Characterization of Invertible Matricesƒƒƒ‚Exercises2.3 Subspaces, Spans, and Linear Independence2.3.1 Subspaces2.3.2 The Span of a Set of Vectors2.3.3 Linear Dependence and Independenceƒƒƒ‚Exercises2.4 Bases and Rank2.4.1 Bases and the Dimension of Subspaces2.4.2 The Fundamental Matrix Subspaces2.4.3 The Rank of a Matrixƒƒƒ‚Exercises2.5 Summary and Review2.A Extra Topic: Linear Algebra Over Finite Fields2.B Extra Topic: Linear Programming2.C Extra Topic: More About the Rank2.D Extra Topic: The LU DecompositionChapter 3: Unraveling Matrices3.1 Coordinate Systems3.1.1 Representations of Vectors3.1.2 Change of Basis3.1.3 Similarity and Representations of Linear Transformationsƒƒƒ‚Exercises3.2 Determinants3.2.1 Definition and Basic Properties3.2.2 Computation3.2.3 Explicit Formulas and Cofactor Expansionsƒƒƒ‚Exercises3.3 Eigenvalues and Eigenvectors3.3.1 Computation of Eigenvalues and Eigenvectors3.3.2 The Characteristic Polynomial and Algebraic Multiplicity3.3.3 Eigenspaces and Geometric Multiplicityƒƒƒ‚Exercises3.4 Diagonalization3.4.1 How to Diagonalize3.4.2 Matrix Powers3.4.3 Matrix Functionsƒƒƒ‚Exercises3.5 Summary and Review3.A Extra Topic: More About Determinants3.B Extra Topic: Power Iteration3.C Extra Topic: Complex Eigenvalues of Real Matrices3.D Extra Topic: Linear Recurrence RelationsAppendix A: Mathematical PreliminariesA.1ƒComplex Numbers A.1.1ƒBasic Arithmetic and Geometry A.1.2ƒThe Complex Conjugate A.1.3ƒEuler’s Formula and Polar FormA.2ƒPolynomials A.2.1ƒRoots of Polynomials A.2.2ƒPolynomial Long Division and the Factor Theorem A.2.3ƒThe Fundamental Theorem of AlgebraA.3ƒProof Techniques A.3.1ƒThe Contrapositive A.3.2ƒBi-Directional Proofs A.3.3ƒProof by Contradiction A.3.4ƒProof by InductionAppendix B: Additional ProofsB.1ƒBlock Matrix MultiplicationB.2ƒUniqueness of Reduced Row Echelon FormB.3ƒMultiplication by an Elementary MatrixB.4ƒExistence of the DeterminantB.5ƒMultiplicity in the Perron–Frobenius TheoremB.6ƒMultiple Roots of PolynomialsB.7ƒLimits of Ratios of Polynomials and ExponentialsAppendix C: Selected Exercise SolutionsC.1ƒChapter 1: Vectors and GeometryC.2ƒChapter 2: Linear Systems and SubspacesC.3ƒChapter 3: Unraveling MatricesBibliographyIndexSymbol Index
1 Vectors and Geometry
	1.1 Vectors and Vector Operations
		1.1.1 Vector Addition
		1.1.2 Scalar Multiplication
		1.1.3 Linear Combinations
		Exercises
	1.2 Lengths, Angles, and the Dot Product
		1.2.1 The Dot Product
		1.2.2 Vector Length
		1.2.3 The Angle Between Vectors
		Exercises
	1.3 Matrices and Matrix Operations
		1.3.1 Matrix Addition and Scalar Multiplication
		1.3.2 Matrix Multiplication
		1.3.3 The Transpose
		1.3.4 Block Matrices
		Exercises
	1.4 Linear Transformations
		1.4.1 Linear Transformations as Matrices
		1.4.2 A Catalog of Linear Transformations
		1.4.3 Composition of Linear Transformations
		Exercises
	1.5 Summary and Review
		Exercises
	1.A Extra Topic: Areas, Volumes, and the Cross Product
		1.A.1 Areas
		1.A.2  Volumes
		Exercises
	1.B Extra Topic: Paths in Graphs
		1.B.1  Undirected Graphs
		1.B.2  Directed Graphs and Multigraphs
		Exercises
2 Linear Systems and Subspaces
	2.1 Systems of Linear Equations
		2.1.1 Matrix Equations
		2.1.2 Row Echelon Form
		2.1.3 Gaussian Elimination
		2.1.4 Solving Linear Systems
		2.1.5 Applications of Linear Systems
		Exercises
	2.2 Elementary Matrices and Matrix Inverses
		2.2.1 Elementary Matrices
		2.2.2 The Inverse of a Matrix
		2.2.3 A Characterization of Invertible Matrices
		Exercises
	2.3 Subspaces, Spans, and Linear Independence
		2.3.1 Subspaces
		2.3.2 The Span of a Set of Vectors
		2.3.3 Linear Dependence and Independence
		Exercises
	2.4 Bases and Rank
		2.4.1 Bases and the Dimension of Subspaces
		2.4.2 The Fundamental Matrix Subspaces
		2.4.3 The Rank of a Matrix
		Exercises
	2.5 Summary and Review
		Exercises
	2.A Extra Topic: Linear Algebra Over Finite Fields
		2.A.1 Binary Linear Systems
		2.A.2 The ``Lights Out'' Game
		2.A.3 Linear Systems with More States
		Exercises
	2.B Extra Topic: Linear Programming
		2.B.1 The Form of a Linear Program
		2.B.2 Geometric Interpretation
		2.B.3 The Simplex Method for Solving Linear Programs
		2.B.4 Duality
		Exercises
	2.C Extra Topic: More About the Rank
		2.C.1 The Rank Decomposition
		2.C.2 Rank in Terms of Submatrices
		Exercises
	2.D Extra Topic: The LU Decomposition
		2.D.1 Computing an LU Decomposition
		2.D.2 Solving Linear Systems
		2.D.3 The PLU Decomposition
		2.D.4 Another Characterization of the LU Decomposition
		Exercises
3 Unraveling Matrices
	3.1 Coordinate Systems
		3.1.1 Representations of Vectors
		3.1.2 Change of Basis
		3.1.3 Similarity and Representations of Linear Transformations
		Exercises
	3.2 Determinants
		3.2.1 Definition and Basic Properties
		3.2.2 Computation
		3.2.3 Explicit Formulas and Cofactor Expansions
		Exercises
	3.3 Eigenvalues and Eigenvectors
		3.3.1 Computation of Eigenvalues and Eigenvectors
		3.3.2 The Characteristic Polynomial and Algebraic Multiplicity
		3.3.3 Eigenspaces and Geometric Multiplicity
		Exercises
	3.4 Diagonalization
		3.4.1 How to Diagonalize
		3.4.2 Matrix Powers
		3.4.3 Matrix Functions
		Exercises
	3.5 Summary and Review
		Exercises
	3.A Extra Topic: More About Determinants
		3.A.1 The Cofactor Matrix and Inverses
		3.A.2 Cramer's Rule
		3.A.3 Permutations
		Exercises
	3.B Extra Topic: Power Iteration
		3.B.1 The Method
		3.B.2 When it Does (and Does Not) Work
		3.B.3 Positive Matrices and Ranking Algorithms
		Exercises
	3.C Extra Topic: Complex Eigenvalues of Real Matrices
		3.C.1 Geometric Interpretation for 2 times2 Matrices
		3.C.2 Block Diagonalization of Real Matrices
		Exercises
	3.D Extra Topic: Linear Recurrence Relations
		3.D.1 Solving via Matrix Techniques
		3.D.2 Directly Solving When Roots are Distinct
		3.D.3 Directly Solving When Roots are Repeated
		Exercises
Appendix A:  Mathematical Preliminaries
	A.1 Complex Numbers
		A.1.1  Basic Arithmetic and Geometry
		A.1.2  The Complex Conjugate
		A.1.3 Euler's Formula and Polar Form
	A.2 Polynomials
		A.2.1  Roots of Polynomials
		A.2.2  Polynomial Long Division and the Factor Theorem
		A.2.3  The Fundamental Theorem of Algebra
	A.3 Proof Techniques
		A.3.1  The Contrapositive
		A.3.2  Bi-Directional Proofs
		A.3.3  Proof by Contradiction
		A.3.4  Proof by Induction
Appendix B:  Additional Proofs
	B.1 Block Matrix Multiplication
	B.2 Uniqueness of Reduced Row Echelon Form
	B.3 Multiplication by an Elementary Matrix
	B.4 Existence of the Determinant
	B.5 Multiplicity in the Perron–Frobenius Theorem
	B.6 Multiple Roots of Polynomials
	B.7 Limits of Ratios of Polynomials and Exponentials
Appendix C:  Selected Exercise Solutions
Bibliography
Index
	Symbol Index