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269 pages

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English

This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms.

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Description

Content

This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms. At the end, the more abstract notions of vector spaces and linear transformations on vector spaces are presented. This is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra.

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Kenneth Kuttler received his Ph.D. in mathematics from The University of Texas at Austin in 1981. From there, he went to Michigan Tech. University where he was employed for most of the next 17 years. He joined the faculty of Brigham Young University in 1998 and has been there since this time. Kuttler's research interests are mainly in the mathematical theory for nonlinear initial boundary value problems, especially those which come from physical models that include damage, contact, and friction. Recently he has become interested in stochastic integration and the related problems involving nonlinear stochastic evolution equations.

- A Few Factorizations
- Definition of an LU Factorization
- Finding an LU Factorization by Inspection
- Using Multipliers To Find An LU Factorization
- Solving Systems Using An LU Factorization
- Justification For The Multiplier Method
- The PLU Factorization
- The QR Factorization
- MATLAB And Factorizations
- Exercises

- Linear Programming
- Simple Geometric Considerations
- The Simplex Tableau
- The Simplex Algorithm
- Finding A Basic Feasible Solution
- Duality
- Exercises

- Spectral Theory
- Eigenvalues And Eigenvectors Of A Matrix
- Some Applications of Eigenvalues and Eigenvectors
- The Estimation Of Eigenvalues
- MATLAB And Eigenvalues
- Exercises

- Matrices And The Inner Product
- Symmetric And Orthogonal Matrices
- Fundamental Theory And Generalizations
- Least Square Approximation
- The Right Polar Factorization
- The Singular Value Decomposition
- Approximation In The Frobenius Norm
- Moore Penrose Inverse
- MATLAB And Singular Value Decomposition
- Exercises

- Numerical Methods For Solving Linear Systems
- Iterative Methods For Linear Systems
- Using MATLAB To Iterate
- The Operator Norm*
- The Condition Number*
- Exercises

- Numerical Methods For Solving The Eigenvalue Problem
- The Power Method For Eigenvalues
- The Shifted Inverse Power Method
- Automation With MATLAB
- The Rayleigh Quotient
- The Algorithm
- MATLAB And The QR Algorithm
- Exercises

- Bibliography
- Appendix C Answers To Selected Exercises
- C.10 Exercises 319
- C.11 Exercises 352
- C.12 Exercises 396
- C.13 Exercises 449
- C.14 Exercises 471
- C.15 Exercises 501

- Endnotes
- Index