Elementary Linear Algebra, Third Edition

Andrilli and Hecker

Table of Contents

Chapter 1: Vectors and Matrices

    Section 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

    Section 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

    Section 1.3: An Introduction to Proof Techniques

        Proof Technique: Direct Proof, Working "Backward" to Discover a Proof, "If A Then B" Proofs, "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 Induction, Negating Statements with Quantifiers and Connectives, Disproving Statements

    Section 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

    Section 1.5: Matrix Multiplication

        Definition of Matrix Multiplication, Application: Shipping Cost and Profit, Fundamental Properties of Matrix Multiplication, Powers of Square Matrices, The Transpose of a Matrix Product

Chapter 2: Systems of Linear Equations

    Section 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) Row Operations, Skipping a Column, Inconsistent Systems, Infinite Solution Sets, Application: Curve Fitting, The Effect of Row Operations on Matrix Multiplication

    Section 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

    Section 2.3: Equivalent Systems, Rank, and Row Space

        Equivalent Systems and Row Equivalence of Matrices, Rank of a Matrix, Homogeneous Systems and Rank, Linear Combinations of Vectors, The Row Space of a Matrix, Row Equivalence Determines the Row Space

    Section 2.4: Inverses of Matrices

        Multiplicative Inverse of a Matrix, Properties of the Matrix Inverse, Inverses for 2×2 matrices, Inverses of Larger Matrices, Using Row Reduction to Show That a Matrix is Singular, Justification of the Inverse Method, Solving a System Using the Inverse of the Coefficient Matrix

Chapter 3: Determinants and Eigenvalues

    Section 3.1: Introduction to Determinants

        Determinants of 1×1, 2×2, and 3×3 Matrices, Application: Areas and Volumes, Cofactors, Formal Definition of the Determinant

    Section 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

    Section 3.3: Further Properties of the Determinant

        Determinant of a Matrix Product, Determinant of the Transpose, A More General Cofactor Expansion, The Adjoint Matrix, Calculating Inverses with the Adjoint Matrix, Cramer's Rule

    Section 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

    Summary of Techniques

Chapter 4: Finite Dimensional Vector Spaces

    Section 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

    Section 4.2: Subspaces

        Definition of a Subspace and Examples, When is a Subset a Subspace?, Checking for Subspaces in Mnn and Rn, Linear Combinations Remain in a Subspace, An Eigenspace is a Subspace

    Section 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, A Spanning Set for an Eigenspace, Special Case: The Span of the Empty Set

    Section 4.4: Linear Independence

        Linear Independence and Dependence, An Alternate Characterization of Linear Independence, An Algebraic Test for Linear Independence, Using Row Reduction to Test for Linear Independence, Linear Independence of Infinite Sets, Uniqueness of Expression of a Vector as a Linear Combination, Linear Independence of Eigenvectors, Summary of Results

    Section 4.5: Basis and Dimension

        Definition of Basis, Two Technical Lemmas, Dimension, Sizes of Spanning Sets and Linearly Independent Sets, Dimension of a Subspace

    Section 4.6: Constructing Special Bases

        Using Row Reduction to Construct a Basis, Every Spanning Set for a Finite Dimensional Vector Space Contains a Basis, Shrinking a Spanning Set to a Basis Using Row Reduction, Shrinking an Infinite Spanning Set to a Basis, Finding a Basis from a Spanning Set by Inspection, Every Linearly Independent Set in a Finite Dimensional Vector Space Is Contained in Some Basis

    Section 4.7: Coordinatization

        Coordinates with Respect to an Ordered Basis, Using Row Reduction to Coordinatize a Vector, Transition Matrix for Change of Coordinates, Calculating the Transition Matrix by Row Reduction, Composition of Transitions, Reversing the Order of Transition, Diagonalization and the Transition Matrix

Chapter 5: Linear Transformations

    Section 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

    Section 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, Similar Matrices, Matrix for the Composition of Linear Transformations

    Section 5.3: The Dimension Theorem

        Kernel and Range, The Dimension Theorem, Finding the Kernel from the Matrix for a Linear Transformation, Finding the Range from the Matrix for a Linear Transformation, Rank of the Transpose

    Section 5.4: Isomorphism

        One-to-One and Onto Linear Transformations, The Kernel of a One-to-One Linear Transformation, Isomorphisms: Invertible Linear Transformations, Isomorphic Vector Spaces, All n-Dimensional Vector Spaces Are Isomorphic, Vector Space Properties Inherited Through Isomorphism

    Section 5.5: Diagonalization of Linear Operators

        Eigenvalues, Eigenvectors, and Eigenspaces for Linear Operators, The Characteristic Polynomial of a Linear Operator, Criterion for Diagonalization, Linear Independence of Eigenvectors, Method for Diagonalizing a Linear Operator, Geometric and Algebraic Multiplicity, Multiplicities and Diagonalization, The Cayley-Hamilton Theorem

Chapter 6: Orthogonality

    Section 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

    Section 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

    Section 6.3: Orthogonal Diagonalization

        Symmetric Operators, A Symmetric Operator Always Has an Eigenvalue, Orthogonally Diagonalizable Operators, Equivalence of Symmetric and Orthogonally Diagonalizable Operators, Method for Orthogonally Diagonalizing a Linear Operator

Chapter 7: Complex Vector Spaces and General Inner Products

    Section 7.1: Complex n-Vectors and Matrices

        Complex n-Vectors, Complex Matrices, Hermitian, Skew-Hermitian, and Normal Matrices

    Section 7.2: Complex Eigenvalues and Eigenvectors

        Complex Linear Systems and Determinants, Complex Eigenvalues and Complex Eigenvectors, Diagonalizable Complex Matrices, Algebraic Multiplicity of an Eigenvalue, Nondiagonalizable Complex Matrices

    Section 7.3: Complex Vector Spaces

        Complex Vector Spaces, Linear Transformations

    Section 7.4: Orthogonality in Cn

        Orthogonal Bases and the Gram-Schmidt Process, Unitary Matrices, Unitarily Diagonalizable Matrices, Self-Adjoint Operators and Hermitian Matrices

    Section 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

Chapter 8: Additional Applications

    Section 8.1: Graph Theory

        Graphs and Digraphs, The Adjacency Matrix, Paths in a Graph or Digraph, Counting Paths

    Section 8.2: Ohm's Law

        Circuit Fundamentals and Ohm's Law

    Section 8.3: Least-Squares Polynomials

        Least-Squares Polynomials

    Section 8.4: Markov Chains

        An Introductory Example, Formal Definitions, Limit Vectors and Fixed Points, Regular Transition Matrices

    Section 8.5: Hill Substitution: An Introduction to Coding Theory

        Substitution Ciphers, Hill Substitution

    Section 8.6: Change of Variables and the Jacobian

        Substitution in One Variable, Double Integrals, Polar Coordinates, Triple Integrals, Spherical Coordinates, Cylindrical Coordinates, Higher Dimensions

    Section 8.7: Rotation of Axes

        Simplifying the Equation of a Conic Section

    Section 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

    Section 8.9: Differential Equations

        First-Order Linear Homogeneous Systems, Higher-Order Homogeneous Differential Equations

    Section 8.10: Least-Squares Solutions for Inconsistent Systems

        Finding Approximate Solutions, Non-unique Least-Squares Solutions, Approximate Eigenvalues and Eigenvectors, Least-Squares Polynomials

    Section 8.11: Max-Min Problems in Rn and the Hessian Matrix

        Taylor's Theorem in Rn, Critical Points, Sufficient Conditions for Local Extreme Points, Positive Definite Quadratic Forms, Local Maxima and Minima in R2, An Example in R3, Failure of the Hessian Matrix Test

Chapter 9: Numerical Methods

    Section 9.1: Numerical Methods for Solving Systems

        Computational Accuracy, Ill-Conditioned Systems, Partial Pivoting, Iterative Techniques: Jacobi and Gauss-Seidel Methods, Comparing Iterative and Row Reduction Methods

    Section 9.2: LDU Decomposition

        Calculating the LDU Decomposition, Solving a System Using LDU Decomposition

    Section 9.3: The Power Method for Finding Eigenvalues

        The Power Method, Problems with the Power Method

Chapter 10: Further Horizons

    Section 10.1: Elementary Matrices

        Using Elementary Matrices to Show Row Equivalence

    Section 10.2: Function Spaces

        Linear Independence in Function Spaces

    Section 10.3: Quadratic Forms

        Quadratic Forms, Orthogonal Change of Basis, The Principal Axes Theorem

Appendix A: Miscellaneous Proofs

    Proof of Theorem 1.14, Part (1), Proof of Theorem 2.9, Proof of Theorem 3.3, Part (3), Case (2), Proof of Theorem 5.28, Proof of Theorem 6.17

Appendix B: Functions

    Functions: Domain, Codomain, and Range, One-to-One and Onto Functions, Composition and Inverses of Functions

Appendix C: Complex Numbers

Appendix D: Computers and Calculators

    Maple 8, Derive 5, Mathematica 4.2, MATLAB 6.5, TI-86 Graphing Calculator, TI-92, TI-92 Plus, TI-89 and Voyage 200 Graphing Calculators

Appendix E: Answers to Selected Exercises

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