Linear Algebra

Introduction and Features. For the Student . . . and Teacher. Prerequisites. Suggested Sequences. 1. Tuples and Vectors. 1.1. Tuples. 1.2. Vectors. 1.3. Proofs. 1.4. Directed Distances. 1.5. Magnitude. 1.6. Direction. 1.7. Unit and Orthogonal Vectors. 2. Systems of Linear Equations. 2.1. Standard Form. 2.2. Solving Systems. 2.3. Coefficient Matrices. 2.4. Free and Basic Variables. 2.5. Computational Considerations. 2.6. Applications of Linear Algebra. 3. Transformations. 3.1. Geometric Transformations. 3.2. Vector Transformations. 3.3. The Transformation Matrix. 3.4. Domain, Codomain, and Range. 3.5. Discrete Time Models. 3.6. Linear Transformations. 3.7. Transformation Arithmetic. 3.8. Cryptography. 4. Matrix Algebra. 4.1. Scalar Multiplication. 4.2. Matrix Addition. 4.3. Matrix Multiplication. 4.4. Elementary Matrices. 4.5. More Transformations. 4.6. Matrix Inverses. 4.7. Complex Matrices. 5. Vector Spaces. 5.1. Vector Spaces. 5.2. Kernels and Null Spaces. 5.3. Span. 5.4. Linear Independence and Dependence. 5.5. Change of Basis. 5.6. Orthogonal Bases. 5.7. Normed Vector Spaces. 5.8. Inner Product Spaces. 5.9. Applications. 5.10. Least Squares. 6. Determinants. 6.1. Linear Equations. 6.2. Transformations. 6.3. Inverse. 6.4. The Determinant. 6.5. A Formula for the Determinant. 6.6. The Determinant Formula. 6.7. More Properties of the Determinant. 6.8. More Computations of the Determinant. 6.9. Use(lesses) of the Determinant. 6.10. Uses of the Determinant. 6.11. Permutations. 7. Eigenvalues and Eigenvectors. 7.1. More Transformations. 7.2. The Eigenproblem. 7.3. Finding Eigenvalues: Numerical Methods. 7.4. Eigenvalues and Eigenvectors for a 2 x 2 Matrix. 7.5. The Characteristic Equation. 7.6. Stochastic Matrices. 7.7. A Determinant-Free Approach. 7.8. Generalized Eigenvalues. 7.9. Symmetric Matrices. 7.10. Graphs. 8. Decomposition. 8.1. LU-Decomposition. 8.2. QR-Decomposition. 8.3. Eigendecompositions. 8.4. Singular Value Decomposition. 9. Extras. 9.1. Properties of Polynomials. 9.2. Complex Numbers. 9.3. Mod-N Arithmetic. 9.4. Polar Coordinates. Bibliography. Index.