Elementary Linear Algebra
Publisher: Bookboon.com
Published: 2013
ISBN-13: 9788740300185
Table of contents: title: Content; title: 1. Some Prerequisite Topics; title: 1.1. Sets And Set Notation; title: 1.2. Functions; title: 1.3. Graphs Of Functions; title: 1.4. The Complex Numbers; title: 1.5. Polar Form Of Complex Numbers; title: 1.6. Roots Of Complex Numbers; title: 1.7. The Quadratic Formula; title: 1.8. Exercises; title: 2. Fn; title: 2.1. Algebra in Fn; title: 2.2. Geometric Meaning Of Vectors; title: 2.3. Geometric Meaning Of Vector Addition; title: 2.4. Distance Between Points In Rn Length Of A Vector; title: 2.5. Geometric Meaning Of Scalar Multiplication; title: 2.6. Exercises; title: 2.7. Vectors And Physics; title: 2.8. Exercises; title: 3. Vector Products; title: 3.1. The Dot Product; title: 3.2. The Geometric Significance Of The Dot Product; title: 3.3. Exercises; title: 3.4. The Cross Product; title: 3.5. The Vector Identity Machine; title: 3.6. Exercises; title: 4. Systems Of Equations; title: 4.1. Systems Of Equations, Geometry; title: 4.2. Systems Of Equations, Algebraic Procedures; title: 4.3. Exercises; title: 5. Matrices; title: 5.1. Matrix Arithmetic; title: 5.2. Exercises; title: 6. Determinants; title: 6.1. Basic Techniques And Properties; title: 6.2. Applications; title: 6.3. Exercises; title: 7. The Mathematical Theory Of Determinants; title: 7.1. The Function sgnn; title: 7.2. The Determinant; title: 7.3. The Cayley Hamilton Theorem; title: 8. Rank Of A Matrix; title: 8.1. Elementary Matrices; title: 8.2. THE Row Reduced Echelon Form Of A Matrix; title: 8.3. The Rank Of A Matrix; title: 8.4. Fredholm Alternative; title: 8.5. Exercises; title: 9. Linear Transformations; title: 9.1. Linear Transformations; title: 9.2. Constructing The Matrix Of A Linear Transformation; title: 9.3. Exercises; title: 10. The LU Factorization; title: 10.1. Definition Of An LU factorization; title: 10.2. Finding An LU Factorization By Inspection; title: 10.3. Using Multipliers To Find An LU Factorization; title: 10.4. Solving Systems Using The LU Factorization; title: 10.5. Justification For The Multiplier Method; title: 10.6. The PLU Factorization; title: 10.7. The QR Factorization; title: 10.8. Exercises; title: 11. Linear Programming; title: 11.1. Simple Geometric Considerations; title: 11.2. The Simplex Tableau; title: 11.3. The Simplex Algorithm; title: 11.4. Finding A Basic Feasible Solution; title: 11.5. Duality; title: 11.6. Exercises; title: 12. Spectral Theory; title: 12.1. Eigenvalues And Eigenvectors Of A Matrix; title: 12.2. Some Applications Of Eigenvalues And Eigenvectors; title: 12.3. The Estimation Of Eigenvalues; title: 12.4. Exercises; title: 13. Matrices And The Inner Product; title: 13.1. Symmetric And Orthogonal Matrices; title: 13.2. Fundamental Theory And Generalizations; title: 13.3. Least Square Approximation; title: 13.4. The Right Polar Factorization; title: 13.5. The Singular Value Decomposition; title: 13.6. Approximation In The Frobenius Norm; title: 13.7. Moore Penrose Inverse; title: 13.8. Exercises; title: 14. Numerical Methods For Solving Linear Systems; title: 14.1. Iterative Methods For Linear Systems; title: 14.2. The Operator Norm; title: 14.3. The Condition Number; title: 14.4. Exercises; title: 15. Numerical Methods For Solving The Eigenvalue Problem; title: 15.1. The Power Method For Eigenvalues; title: 15.2. The Shifted Inverse Power Method; title: 15.3. The Rayleigh Quotient; title: 15.4. The QR Algorithm; title: 15.5. Exercises; title: 15.6. Vector Spaces; title: 15.6.1. Algebraic Considerations; title: 15.6.2. Exercises; title: 15.6.3. Vector Spaces And Fields; title: 15.6.4. Exercises; title: 15.6.5. Inner Product Spaces; title: 15.6.6. Exercises; title: 15.7. Linear Transformations; title: 15.7.1. Matrix Multiplication As A Linear Transformation; title: 15.7.2. L(V,W) As A Vector Space; title: 15.7.3. Eigenvalues And Eigenvectors Of Linear Transformations; title: 15.7.4. Block Diagonal Matrices; title: 15.7.5. The Matrix Of A Linear Transformation; title: 15.7.6. Exercises; title: 15.8. The Jordan Canonical Form*; title: 15.9. The Fundamental Theorem Of Algebra; title: 15.10. Answers To Selected Exercises