Elementary Linear Algebra (Sixth Edition)

Table of Contents Preface for the Instructor Preface to the Student A Light-Hearted Look at Linear Algebra Terms Symbol Table Computational & Numerical Techniques, Applications 1 Vectors and Matrices 1.1 Fundamental Operations With Vectors Definition of a Vector Geometric Interpretation of Vectors Length of a Vector Scalar Multiplication and Parallel Vectors Addition and Subtraction With Vectors Fundamental Properties of Addition and Scalar Multiplication Linear Combinations of Vectors Physical Applications of Addition and Scalar Multiplication New Vocabulary Highlights Exercises for Section 1.1 1.2 The Dot Product Definition and Properties of the Dot Product Inequalities Involving the Dot Product The Angle Between Two Vectors Special Cases: Orthogonal and Parallel Vectors Projection Vectors Application: Work New Vocabulary Highlights Exercises for Section 1.2 1.3 An Introduction to Proof Techniques Proof Technique: Direct Proof ``If A Then B'' Proofs Working ``Backward'' to Discover a Proof ``A If and Only If B'' Proofs ``If A Then (B or C)'' Proofs Proof Technique: Proof by Contrapositive Converse and Inverse Proof Technique: Proof by Contradiction Proof Technique: Proof by Induction Proof Technique: Reducing to a Previous Result Negating Statements With Quantifiers and Connectives Disproving Statements New Vocabulary Highlights Exercises for Section 1.3 1.4 Fundamental Operations With Matrices Definition of a Matrix Special Types of Matrices Addition and Scalar Multiplication With Matrices Fundamental Properties of Addition and Scalar Multiplication The Transpose of a Matrix and Its Properties Symmetric and Skew-Symmetric Matrices New Vocabulary Highlights Exercises for Section 1.4 1.5 Matrix Multiplication Definition of Matrix Multiplication Application: Shipping Cost and Profit Linear Combinations Using Matrix Multiplication Fundamental Properties of Matrix Multiplication Powers of Square Matrices The Transpose of a Matrix Product New Vocabulary Highlights Exercises for Section 1.5 Review Exercises for Chapter 1 2 Systems of Linear Equations 2.1 Solving Linear Systems Using Gaussian Elimination Systems of Linear Equations Number of Solutions to a System Gaussian Elimination Row Operations and Their Notation The Strategy in the Simplest Case Using Type (III) Operations Skipping a Column Inconsistent Systems Infinite Solution Sets Application: Curve Fitting The Effect of Row Operations on Matrix Multiplication New Vocabulary Highlights Exercises for Section 2.1 2.2 Gauss-Jordan Row Reduction and Reduced Row Echelon Form Introduction to Gauss-Jordan Row Reduction Reduced Row Echelon Form Number of Solutions Homogeneous Systems Application: Balancing Chemical Equations Solving Several Systems Simultaneously New Vocabulary Highlights Exercises for Section 2.2 2.3 Equivalent Systems, Rank, and Row Space Equivalent Systems and Row Equivalence of Matrices Rank of a Matrix Linear Combinations of Vectors The Row Space of a Matrix Row Equivalence Determines the Row Space New Vocabulary Highlights Exercises for Section 2.3 2.4 Inverses of Matrices Multiplicative Inverse of a Matrix Properties of the Matrix Inverse Inverses for 2x2 Matrices Inverses of Larger Matrices Justification of the Inverse Method Solving a System Using the Inverse of the Coefficient Matrix New Vocabulary Highlights Exercises for Section 2.4 Review Exercises for Chapter 2 3 Determinants and Eigenvalues 3.1 Introduction to Determinants Determinants of 1x1, 2x2, and 3x3 Matrices Application: Areas and Volumes Cofactors Formal Definition of the Determinant New Vocabulary Highlights Exercises for Section 3.1 3.2 Determinants and Row Reduction Determinants of Upper Triangular Matrices Effect of Row Operations on the Determinant Calculating the Determinant by Row Reduction Determinant Criterion for Matrix Singularity Highlights Exercises for Section 3.2 3.3 Further Properties of the Determinant Determinant of a Matrix Product Determinant of the Transpose A More General Cofactor Expansion Cramer's Rule New Vocabulary Highlights Exercises for Section 3.3 3.4 Eigenvalues and Diagonalization Eigenvalues and Eigenvectors The Characteristic Polynomial of a Matrix Diagonalization Nondiagonalizable Matrices Algebraic Multiplicity of an Eigenvalue Application: Large Powers of a Matrix Roundoff Error Involving Eigenvalues New Vocabulary Highlights Exercises for Section 3.4 Review Exercises for Chapter 3 Summary of Techniques Techniques for Solving a System AX=B of m Linear Equations in n Unknowns Techniques for Finding the Inverse (If It Exists) of an nxn Matrix A Techniques for Finding the Determinant of an nxn Matrix A Techniques for Finding the Eigenvalues of an nxn Matrix A Technique for Finding the Eigenvectors of an nxn Matrix A 4 Finite Dimensional Vector Spaces 4.1 Introduction to Vector Spaces Definition of a Vector Space Examples of Vector Spaces Two Unusual Vector Spaces Some Elementary Properties of Vector Spaces Failure of the Vector Space Conditions New Vocabulary Highlights Exercises for Section 4.1 4.2 Subspaces Definition of a Subspace and Examples When Is a Subset a Subspace? Verifying Subspaces in Mnn and Rn Subsets That Are Not Subspaces Linear Combinations Remain in a Subspace An Eigenspace Is a Subspace New Vocabulary Highlights Exercises for Section 4.2 4.3 Span Finite Linear Combinations Definition of the Span of a Set Span(S) Is the Minimal Subspace Containing S Simplifying Span(S) Using Row Reduction Special Case: The Span of the Empty Set New Vocabulary Highlights Exercises for Section 4.3 4.4 Linear Independence Linear Independence and Dependence Linear Dependence and Independence With One- and Two-Element Sets Using Row Reduction to Test for Linear Independence Alternate Characterizations of Linear Independence Linear Independence of Infinite Sets Uniqueness of Expression of a Vector as a Linear Combination Summary of Results New Vocabulary Highlights Exercises for Section 4.4 4.5 Basis and Dimension Definition of Basis A Technical Lemma Dimension Sizes of Spanning Sets and Linearly Independent Sets Dimension of a Subspace Diagonalization and Bases New Vocabulary Highlights Exercises for Section 4.5 4.6 Constructing Special Bases Using the Simplified Span Method to Construct a Basis Using the Independence Test Method to Shrink a Spanning Set to a Basis Enlarging a Linearly Independent Set to a Basis New Vocabulary Highlights Exercises for Section 4.6 4.7 Coordinatization Coordinates With Respect to an Ordered Basis Using Row Reduction to Coordinatize a Vector Fundamental Properties of Coordinatization The Transition Matrix for Change of Coordinates Change of Coordinates Using the Transition Matrix Algebra of the Transition Matrix Diagonalization and the Transition Matrix New Vocabulary Highlights Exercises for Section 4.7 Review Exercises for Chapter 4 5 Linear Transformations 5.1 Introduction to Linear Transformations Functions Linear Transformations Linear Operators and Some Geometric Examples Multiplication Transformation Elementary Properties of Linear Transformations Linear Transformations and Subspaces New Vocabulary Highlights Exercises for Section 5.1 5.2 The Matrix of a Linear Transformation A Linear Transformation Is Determined by Its Action on a Basis The Matrix of a Linear Transformation Finding the New Matrix for a Linear Transformation After a Change of Basis Linear Operators and Similarity Matrix for the Composition of Linear Transformations New Vocabulary Highlights Exercises for Section 5.2 5.3 The Dimension Theorem Kernel and Range Finding the Kernel From the Matrix of a Linear Transformation Finding the Range From the Matrix of a Linear Transformation The Dimension Theorem Rank of the Transpose New Vocabulary Highlights Exercises for Section 5.3 5.4 One-to-One and Onto Linear Transformations One-to-One and Onto Linear Transformations Characterization by Kernel and Range Spanning and Linear Independence New Vocabulary Highlights Exercises for Section 5.4 5.5 Isomorphism Isomorphisms: Invertible Linear Transformations Isomorphic Vector Spaces Isomorphism of n-Dimensional Vector Spaces Isomorphism and the Methods Proving the Dimension Theorem Using Isomorphism New Vocabulary Highlights Exercises for Section 5.5 5.6 Diagonalization of Linear Operators Eigenvalues, Eigenvectors, and Eigenspaces for Linear Operators The Characteristic Polynomial of a Linear Operator Criteria for Diagonalization Linear Independence of Eigenvectors Method for Diagonalizing a Linear Operator Geometric and Algebraic Multiplicity Multiplicities and Diagonalization The Cayley-Hamilton Theorem New Vocabulary Highlights Exercises for Section 5.6 Review Exercises for Chapter 5 6 Orthogonality 6.1 Orthogonal Bases and the Gram-Schmidt Process Orthogonal and Orthonormal Vectors Orthogonal and Orthonormal Bases The Gram-Schmidt Process: Finding an Orthogonal Basis for a Subspace of Rn Orthogonal Matrices New Vocabulary Highlights Exercises for Section 6.1 6.2 Orthogonal Complements Orthogonal Complements Properties of Orthogonal Complements Orthogonal Projection Onto a Subspace Application: Orthogonal Projections and Reflections in R3 Application: Distance From a Point to a Subspace New Vocabulary Highlights Exercises for Section 6.2 6.3 Orthogonal Diagonalization Symmetric Operators Orthogonally Diagonalizable Operators A Symmetric Operator Always Has an Eigenvalue Equivalence of Symmetric and Orthogonally Diagonalizable Operators Method for Orthogonally Diagonalizing a Linear Operator The Spectral Theorem New Vocabulary Highlights Exercises for Section 6.3 Review Exercises for Chapter 6 7 Complex Vector Spaces and General Inner Products 7.1 Complex n-Vectors and Matrices Complex n-Vectors Complex Matrices Hermitian, Skew-Hermitian, and Normal Matrices New Vocabulary Highlights Exercises for Section 7.1 7.2 Complex Eigenvalues and Complex Eigenvectors Complex Linear Systems and Determinants Complex Eigenvalues and Complex Eigenvectors Diagonalizable Complex Matrices and Algebraic Multiplicity Nondiagonalizable Complex Matrices New Vocabulary Highlights Exercises for Section 7.2 7.3 Complex Vector Spaces Complex Vector Spaces Linear Transformations New Vocabulary Highlights Exercises for Section 7.3 7.4 Orthogonality in Cn Orthogonal Bases and the Gram-Schmidt Process Unitary Matrices Unitarily Diagonalizable Matrices Self-Adjoint Operators and Hermitian Matrices New Vocabulary Highlights Exercises for Section 7.4 7.5 Inner Product Spaces Inner Products Length, Distance, and Angles in Inner Product Spaces Orthogonality in Inner Product Spaces The Generalized Gram-Schmidt Process Orthogonal Complements and Orthogonal Projections in Inner Product Spaces New Vocabulary Highlights Exercises for Section 7.5 Review Exercises for Chapter 7 8 Additional Applications 8.1 Graph Theory Graphs and Digraphs The Adjacency Matrix Paths in a Graph or Digraph Counting Paths Connected Graphs New Vocabulary Highlights Exercises for Section 8.1 8.2 Ohm's Law Circuit Fundamentals and Ohm's Law New Vocabulary Highlights Exercises for Section 8.2 8.3 Least-Squares Polynomials Least-Squares Polynomials Least-Squares Lines Least-Squares Quadratics Generalization of the Process New Vocabulary Highlights Exercises for Section 8.3 8.4 Markov Chains An Introductory Example Formal Definitions Limit Vectors and Fixed Points Regular Transition Matrices New Vocabulary Highlights Exercises for Section 8.4 8.5 Hill Substitution: An Introduction to Coding Theory Substitution Ciphers Hill Substitution New Vocabulary Highlights Exercises for Section 8.5 8.6 Linear Recurrence Relations and the Fibonacci Sequence The Fibonacci Sequence A Linear Recurrence Relation of Order 3 Exercises for Section 8.6 8.7 Rotation of Axes for Conic Sections Simplifying the Equation of a Conic Section New Vocabulary Highlights Exercises for Section 8.7 8.8 Computer Graphics Introduction to Computer Graphics Fundamental Movements in the Plane Homogeneous Coordinates Representing Movements With Matrix Multiplication in Homogeneous Coordinates Movements Not Centered at the Origin Composition of Movements New Vocabulary Highlights Exercises for Section 8.8 8.9 Differential Equations First-Order Linear Homogeneous Systems Higher-Order Homogeneous Differential Equations New Vocabulary Highlights Exercises for Section 8.9 8.10 Least-Squares Solutions for Inconsistent Systems Finding Approximate Solutions Non-Unique Least-Squares Solutions Approximate Eigenvalues and Eigenvectors Least-Squares Polynomials New Vocabulary Highlights Exercises for Section 8.10 8.11 Quadratic Forms Quadratic Forms Orthogonal Change of Basis The Principal Axes Theorem New Vocabulary Highlights Exercises for Section 8.11 9 Numerical Techniques 9.1 Numerical Techniques for Solving Systems Computational Accuracy Ill-Conditioned Systems Partial Pivoting Iterative Techniques: Jacobi and Gauss-Seidel Methods Comparing Iterative and Row Reduction Methods New Vocabulary Highlights Exercises for Section 9.1 9.2 LDU Decomposition Calculating the LDU Decomposition Solving a System Using LDU Decomposition New Vocabulary Highlights Exercises for Section 9.2 9.3 The Power Method for Finding Eigenvalues The Power Method Problems With the Power Method New Vocabulary Highlights Exercises for Section 9.3 9.4 QR Factorization QR Factorization Theorem QR Factorization and Least Squares A More General QR Factorization New Vocabulary Highlights Exercises for Section 9.4 9.5 Singular Value Decomposition Singular Values and Right Singular Vectors Singular Values and Left Singular Vectors Orthonormal Bases Derived From the Left and Right Singular Vectors Singular Value Decomposition A Geometric Interpretation The Outer Product Form for Singular Value Decomposition Digital Images The Pseudoinverse New Vocabulary Highlights Exercises for Section 9.5 A Miscellaneous Proofs B Functions Exercises for Appendix B C Complex Numbers Exercises for Appendix C D Elementary Matrices Exercises for Appendix D E Answers to Selected Exercises Index Equivalent Conditions for Singular and Nonsingular Matrices Diagonalization Method Simplified Span Method (Simplifying Span(S)) Independence Test Method (Testing for Linear Independence of S) Equivalent Conditions for Linearly Independent and Linearly Dependent Sets Coordinatization Method (Coordinatizing v with Respect to an Ordered Basis B) Transition Matrix Method (Calculating a Transition Matrix from B to C) Kernel Method (Finding a Basis for the Kernel of L) Range Method (Finding a Basis for the Range of L) Equivalence Conditions for One-to-One, Onto, and Isomorphism Dimension Theorem Gram-Schmidt Process