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’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.
Author(s): Nathaniel Johnston
Publisher: Springer
Year: 2021
Language: English
Pages: 492
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 CombinationsExercises1.2 Lengths, Angles, and the Dot Product1.2.1 The Dot Product1.2.2 Vector Length1.2.3 The Angle Between VectorsExercises1.3 Matrices and Matrix Operations1.3.1 Matrix Addition and Scalar Multiplication1.3.2 Matrix Multiplication1.3.3 The Transpose1.3.4 Block MatricesExercises1.4 Linear Transformations1.4.1 Linear Transformations as Matrices1.4.2 A Catalog of Linear Transformations1.4.3 Composition of Linear TransformationsExercises1.5 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 SystemsExercises2.2 Elementary Matrices and Matrix Inverses2.2.1 Elementary Matrices2.2.2 The Inverse of a Matrix2.2.3 A Characterization of Invertible MatricesExercises2.3 Subspaces, Spans, and Linear Independence2.3.1 Subspaces2.3.2 The Span of a Set of Vectors2.3.3 Linear Dependence and IndependenceExercises2.4 Bases and Rank2.4.1 Bases and the Dimension of Subspaces2.4.2 The Fundamental Matrix Subspaces2.4.3 The Rank of a MatrixExercises2.5 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 TransformationsExercises3.2 Determinants3.2.1 Definition and Basic Properties3.2.2 Computation3.2.3 Explicit Formulas and Cofactor ExpansionsExercises3.3 Eigenvalues and Eigenvectors3.3.1 Computation of Eigenvalues and Eigenvectors3.3.2 The Characteristic Polynomial and Algebraic Multiplicity3.3.3 Eigenspaces and Geometric MultiplicityExercises3.4 Diagonalization3.4.1 How to Diagonalize3.4.2 Matrix Powers3.4.3 Matrix FunctionsExercises3.5 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.1Complex Numbers A.1.1Basic Arithmetic and Geometry A.1.2The Complex Conjugate A.1.3Euler’s Formula and Polar FormA.2Polynomials A.2.1Roots of Polynomials A.2.2Polynomial Long Division and the Factor Theorem A.2.3The Fundamental Theorem of AlgebraA.3Proof Techniques A.3.1The Contrapositive A.3.2Bi-Directional Proofs A.3.3Proof by Contradiction A.3.4Proof by InductionAppendix B: Additional ProofsB.1Block Matrix MultiplicationB.2Uniqueness of Reduced Row Echelon FormB.3Multiplication by an Elementary MatrixB.4Existence of the DeterminantB.5Multiplicity in the Perron–Frobenius TheoremB.6Multiple Roots of PolynomialsB.7Limits of Ratios of Polynomials and ExponentialsAppendix C: Selected Exercise SolutionsC.1Chapter 1: Vectors and GeometryC.2Chapter 2: Linear Systems and SubspacesC.3Chapter 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
