1. Systems of Linear Equations
1.1. Introduction to Systems of Linear Equations
- Linear Equations in n Variables
- Solutions and Solution Sets
- Systems of Linear Equations
- Solving a System of Linear Equations
1.1. Exercises
1.2. Gaussian Elimination and Gauss-Jordan Elimination
- Matrices
- Elementary Row Operations
- Gauss-Jordan Elimination
- Homogeneous Systems of Linear Equations
1.2. Exercises
1.3. Applications of Systems of Linear Equations
- Polynomial Curve Fitting
- Network Analysis
1.3. Exercises
- Review Exercises
- Projects
2. Matrices
2.1. Operations with Matrices
- Equality of Matrices
- Matrix Addition, Subtraction, and Scalar Multiplication
- Matrix Multiplication
- Systems of Linear Equations
- Partitioned Matrices
2.1. Exercises
2.2. Properties of Matrix Operations
- Algebra of Matrices
- Properties of Matrix Multiplication
- The Transpose of a Matrix
2.2. Exercises
2.3. The Inverse of a Matrix
- Matrices and Their Inverses
- Properties of Inverses
- Systems of Equations
2.3. Exercises
2.4. Elementary Matrices
- Elementary Matrices and Elementary Row Operations
- The L U -Factorization
2.4. Exercises
2.5. Markov Chains
- Stochastic Matrices and Markov Chains
- Steady State Matrix of a Markov Chain
- Absorbing Markov Chains
2.5. Exercises
2.6. More Applications of Matrix Operations
- Cryptography
- Leontief Input-Output Models
- Least Squares Regression Analysis
2.6. Exercises
- Review Exercises
- Projects
3. Determinants
3.1. The Determinant of a Matrix
- The Determinant of a 2 × 2 Matrix
- Minors and Cofactors
- The Determinant of a Square Matrix
- Triangular Matrices
3.1. Exercises
3.2. Determinants and Elementary Operations
- Determinants and Elementary Row Operations
- Determinants and Elementary Column Operations
- Matrices and Zero Determinants
3.2. Exercises
3.3. Properties of Determinants
- Matrix Products and Scalar Multiples
- Determinants and the Inverse of a Matrix
- Determinants and the Transpose of a Matrix
3.3. Exercises
3.4. Applications of Determinants
- The Adjoint of a Matrix
- Cramer’s Rule
- Area, Volume, and Equations of Lines and Planes
3.4. Exercises
- Review Exercises
- Projects
1-3. Cumulative Test
4. Vector Spaces
4.1. Vectors in Rⁿ
- Vectors in the Plane
- Vector Operations
- Vectors in Rⁿ
- Linear Combinations of Vectors
4.1. Exercises
4.2. Vector Spaces
- Definition of a Vector Space
- Sets that Are Not Vector Spaces
4.2. Exercises
4.3. Subspaces of Vector Spaces
- Subspaces
- Subspaces of Rⁿ
4.3. Exercises
4.4. Spanning Sets and Linear Independence
- Linear Combinations of Vectors in a Vector Space
- Spanning Sets
- Linear Dependence and Linear Independence
4.4. Exercises
4.5. Basis and Dimension
- Basis for a Vector Space
- The Dimension of a Vector Space
4.5. Exercises
4.6. Rank of a Matrix and Systems of Linear Equations
- Row Space, Column Space, and Rank of a Matrix
- The Nullspace of a Matrix
- Solutions of Systems of Linear Equations
4.6. Exercises
4.7. Coordinates and Change of Basis
- Coordinate Representation in Rⁿ
- Change of Basis in Rⁿ
- Coordinate Representation in General n-Dimensional Spaces
4.7. Exercises
4.8. Applications of Vector Spaces
- Linear Differential Equations (Calculus)
- Conic Sections and Rotation
4.8. Exercises
- Review Exercises
- Projects
5. Inner Product Spaces
5.1. Length and Dot Product in Rⁿ
- Vector Length and Unit Vectors
- Distance Between Two Vectors in Rⁿ
- Dot Product and the Angle Between Two Vectors
- The Dot Product and Matrix Multiplication
5.1. Exercises
5.2. Inner Product Spaces
- Inner Products
- Orthogonal Projections in Inner Product Spaces
5.2. Exercises
5.3. Orthonormal Bases: Gram-Schmidt Process
- Orthogonal and Orthonormal Sets
- Gram-Schmidt Orthonormalization Process
5.3. Exercises
5.4. Mathematical Models and Least Squares Analysis
- The Least Squares Problem
- Orthogonal Subspaces
- Fundamental Subspaces of a Matrix
- Solving the Least Squares Problem
- Mathematical Modeling
5.4. Exercises
5.5. Applications of Inner Product Spaces
- The Cross Product of Two Vectors in R³
- Least Squares Approximations (Calculus)
- Fourier Approximations (Calculus)
5.5. Exercises
- Review Exercises
- Projects
4 and 5 Cumulative Test
6. Linear Transformations
6.1. Introduction to Linear Transformations
- Images and Preimages of Functions
- Linear Transformations
6.1. Exercises
6.2. The Kernel and Range of a Linear Transformation
- The Kernel of a Linear Transformation
- The Range of a Linear Transformation
- One-to-One and Onto Linear Transformations
- Isomorphisms of Vector Spaces
6.2. Exercises
6.3. Matrices for Linear Transformations
- The Standard Matrix for a Linear Transformation
- Composition of Linear Transformations
- Nonstandard Bases and General Vector Spaces
6.3. Exercises
6.4. Transition Matrices and Similarity
- The Matrix for a Linear Transformation
- Similar Matrices
6.4. Exercises
6.5. Applications of Linear Transformations
- The Geometry of Linear Transformations in R²
- Rotation in R³
6.5. Exercises
- Review Exercises
- Projects
7. Eigenvalues and Eigenvectors
7.1. Eigenvalues and Eigenvectors
- The Eigenvalue Problem
- Eigenspaces
- Finding Eigenvalues and Eigenvectors
- Eigenvalues and Eigenvectors of Linear Transformations
7.1. Exercises
7.2. Diagonalization
- The Diagonalization Problem
- Diagonalization and Linear Transformations
7.2. Exercises
7.3. Symmetric Matrices and Orthogonal Diagonalization
- Symmetric Matrices
- Orthogonal Matrices
- Orthogonal Diagonalization
7.3. Exercises
7.4. Applications of Eigenvalues and Eigenvectors
- Population Growth
- Systems of Linear Differential Equations (Calculus)
- Quadratic Forms
- Constrained Optimization
7.4. Exercises
- Review Exercises
- Projects
6 and 7 Cumulative Test
8. Complex Vector Spaces
8.1. Complex Numbers
- Complex Numbers
- The Complex Plane
- Addition, Subtraction, and Scalar Multiplication of Complex Numbers
- Multiplication of Complex Numbers
- Complex Matrices
8.1. Exercises
8.2. Conjugates and Division of Complex Numbers
- Complex Conjugates
- The Modulus of a Complex Number
- Division of Complex Numbers
8.2. Exercises
8.3. Polar Form and DeMoivre’s Theorem
- Polar Form of a Complex Number
- DeMoivre’s Theorem
8.3. Exercises
8.4. Complex Vector Spaces and Inner Products
- Complex Vector Spaces
- Complex Inner Product Spaces
8.4. Exercises
8.5. Unitary and Hermitian Matrices
- Conjugate Transpose of a Matrix
- Unitary Matrices
- Hermitian Matrices
8.5. Exercises
- Review Exercises
- Projects
9. Linear Programming
9.1. Systems of Linear Inequalities
- Linear Inequalities and Their Graphs
- Systems of Inequalities
9.1. Exercises
9.2. Linear Programming Involving Two Variables
- Solving a Linear Programming Problem
- Application
9.2. Exercises
9.3. The Simplex Method: Maximization
- The Simplex Tableau
- Pivoting
- The Simplex Method
- Applications
9.3. Exercises
9.4. The Simplex Method: Minimization
- The Dual of a Minimization Problem
- Solving a Minimization Problem
9.4. Exercises
9.5. The Simplex Method: Mixed Constraints
- Mixed Constraints and Maximization
- Mixed Constraints and Minimization
9.5. Exercises
- Review Exercises
- Project
10. Numerical Methods
10.1. Gaussian Elimination with Partial Pivoting
- Stored Value and Rounding Error
- Gaussian Elimination with Partial Pivoting
10.1. Exercises
10.2. Iterative Methods for Solving Linear Systems
- The Jacobi Method
- The Gauss-Seidel Method
- Convergence and Divergence
10.2. Exercises
10.3. Power Method for Approximating Eigenvalues
- Dominant Eigenvalues and Dominant Eigenvectors
- The Power Method
10.3. Exercises
10.4. Applications of Numerical Methods
- Applications of Gaussian Elimination with Partial Pivoting
- Applications of the Gauss-Seidel Method
- Applications of the Power Method
10.4. Exercises
- Review
- Projects
Appendix. Mathematical Induction and Other Forms of Proofs
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