Preface
Chapter 1 Systems of Linear Equations
1.1 Matrices and Systems of Linear Equations
1.2 Gauss-Jordan Elimination
1.3* Curve Fitting, Electrical Networks, and Traffic Flow
Chapter 1 Review Exercises
Chapter 2 Matrices
2.1 Addition, Scalar Multiplication, and Multiplication of Matrices
2.2 Algebraic Properties of Matrix Operations
2.3 Symmetric Matrices and Seriation in Archaeology
2.4 Inverse of a Matrix, Elementary Matrices, Color Models and Cryptography
2.5* The Leontief Input-Output Model in Economics
2.6* Markov Chains, Population Movements, and Genetics
2.7* A Communication Model and Group Relationships in Sociology
Chapter 2 Review Exercises
Chapter 3 Determinants
3.1 Introduction to Determinants
3.2 Properties of Determinants
3.3* Numerical Evaluation of a Determinant
3.4 Determinants, Matrix Inverses, and Systems of Linear Equations
Chapter 3 Review Exercises
Chapter 4 Vector Spaces
4.1 The Vector Space Rn
4.2 Dot Product, Norm, Angle and Distance
4.3 General Vector Spaces
4.4 Subspaces
4.5 Linear Combinations of Vectors
4.6 Linear Dependence and Independence
4.7 Basis and Dimension
4.8 Rank of a Matrix
4.9 Orthonormal Vectors and Projections in Rn
Chapter 4 Review Exercises
Chapter 5 Eigenvalues and Eigenvectors
5.1 Eigenvalues, Eigenvectors, and Eigenspaces
5.2* Demography and Weather Prediction
5.3 Diagonalization of Matrices
5.4* Quadratic Forms, Difference Equations, and Normal Modes
Chapter 5 Review Exercises
Chapter 6 Linear Transformations
6.1 Introduction to Linear Transformations
6.2 Matrix Transformations, Computer Graphics, and Fractals
6.3 Kernel and Range
6.4* Transformations and Systems of Linear Equations
6.5 Coordinate Vectors
6.6 Matrix Representations of Linear Transformations
Chapter 6 Review Exercises
Chapter 7 Inner Product Spaces
7.1 Inner Product Spaces
7.2* Non-Euclidean Geometry and Special Relativity
7.3* Approximation of Functions and Coding Theory
7.4* Least Squares Curves
Chapter 7 Review Exercises
Chapter 8 Numerical Techniques
8.1* Gaussian Elimination
8.2* The Method of LU Decomposition
8.3* Practical Difficulties in Solving Systems of Equations
8.4* Iterative Methods for Solving Systems of Linear Equations
8.5* Eigenvalues by Iteration and Connectivity of Networks
Chapter 8 Review Exercises
Chapter 9 Linear Programming
9.1* A Geometrical Introduction to Linear Programming
9.2* The Simplex Method
9.3* Geometrical Explanation of the Simplex Method
Chapter 9 Review Exercises
Appendices
A Cross Product
B Equations of Lines in Three-Space
C Graphing Calculator Manual
D MATLAB Manual
Answers to Selected Exercises
Index
Appendix C Graphing Calculator Manual
C1 Solving Systems of Linear Equations in Two Variables Graphically (Section 1.1)
C2 Row Operations (Sections 1.1, 1.2, 1.3)
C3 Addition and Scalar Multiplication of Matrices (Section 2.1)
C4 Multiplication of Matrices and Powers of Matrices (Section 2.1)
C5 Transpose of a Matrix (Section 2.3)
C6 Inverse of a Matrix (Section 2.4)
C7 Determinant of a Matrix (Section 3.1)
Appendix D MATLAB Manual
D1 Entering and Displaying a Matrix (Section 1.1)
D2 Solving Systems of Linear Equations (Sections 1.1, 1.2, 1.3)
D3 Matrix Operations, Submatrices, Rows and Columns (Sections 2.1, 2.2, 2.3)
D4 Computational Considerations (Section 2.2)
D5 Inverse of a Matrix (Section 2.4)
D6 Solving Systems of Equations using Matrix Inverse (Section 2.4)
D7 Cryptography (Section 2.4)
D8 Leontief I/O Model (Section 2.5)
D9 Markov Chains (Sections 2.6, 6.2)
D10 Digraphs (Section 2.7)
D11 Determinants (Sections 3.1, 3.2, 3.3)
D12 Cramer's Rule (Section 3.4)
D13 Dot Product, Norm, Angle, Distance, (Section 4.2)
D14 Linear Combinations, Linear Dependence, Basis (Sections 4.4 - 4.7)
D15 Projection, Gram - Schmidt Orthogonalization (Section 4.8)
D16 Eigenvalues and Eigenvectors (Sections 5.1 - 5.4)
D17 Transformations defined by Matrices (Sections 6.1, 6.2)
D18 Fractals (Section 6.2)
D19 Kernel and Range (Section 6.3)
D20 Inner Product, Non-Euclidean Geometry (Sections 7.1, 7.2)
D21 Space-Time Travel (Section 7.2)
D22 Pseudoinverse and Least Squares Curves (Section 7.4)
D23 LU Decomposition (Section 8.2)
D24 Condition Number of a Matrix (Section 8.3)
D25 Jacobi and Gauss-Seidel Iterative Methods (Section 8.4)
D26 The Simplex Method in Linear Programming (Section 9.2)
D27 Cross Product ( Appendix A )
D28 MATLAB Commands, Functions and M-files
D29 The Linear Algebra with Applications Toolbox M-Files
Note: Chapters and sections with asterisks (*) are optional. The instructor can use these sections to build around the core material to give the course the desired emphasis.
