Introduction to Linear and Matrix Algebra

This textbook emphasizes the interplay between algebra and geometry to motivate the study of linear algebra. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. By focusing on this interface, the author offers a conceptual appreciation of the mathematics that is at the heart of further theory and applications. Those continuing to a second course in linear algebra will appreciate the companion volume Advanced Linear and Matrix Algebra. Starting with an introduction to vectors, matrices, and linear transformations, the book focuses on building a geometric intuition of what these tools represent. Linear systems offer a powerful application of the ideas seen so far, and lead onto the introduction of subspaces, linear independence, bases, and rank. Investigation then focuses on the algebraic properties of matrices that illuminate the geometry of the linear transformations that they represent. Determinants, eigenvalues, and eigenvectors all benefit from this geometric viewpoint. Throughout, ?Extra Topic? sections augment the core content with a wide range of ideas and applications, from linear programming, to power iteration and linear recurrence relations. Exercises of all levels accompany each section, including many designed to be tackled using computer software. Introduction to Linear and Matrix Algebra is ideal for an introductory proof-based linear algebra course. The engaging color presentation and frequent marginal notes showcase the author{u2019}s visual approach. Students are assumed to have completed one or two university-level mathematics courses, though calculus is not an explicit requirement. Instructors will appreciate the ample opportunities to choose topics that align with the needs of each classroom, and the online homework sets that are available through WeBWorK 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