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Transform Linear Algebra

By Frank Uhlig

Published by Pearson

Published Date: Nov 2, 2001

Table of Contents

(NOTE: * Available on Web only).

Introduction (Mathematical Preliminaries, Vectors, Sets, and Symbols).


1. Linear Transformations.

Lecture One: Vectors, Linear Functions, and Matrices. Tasks and Methods of Linear Algebra. Applications: Geometry, Calculus, and MATLAB.



2. Row-Reduction.

Lecture Two: Gaussian Elimination and the Echelon Forms. Applications: MATLAB.



3. Linear Equations.

Lecture Three: Solvability and Solutions of Linear Systems. Applications: Circuits, Networks, Chemistry, and MATLAB.



4. Subspaces.

Lecture Four: The Image and Kernel of a Linear Transformation. Applications: Join and Intersection of Subspaces.



5. Linear Dependence, Bases, and Dimension.

Lecture Five: Minimal Spanning or Maximally Independent Sets of Vectors. Applications: Multiple Spanning Sets of One Subspace, MATLAB.



6. Composition of Maps, Matrix Inverse.

Lecture Six. Theory: Gauss Elimination Matrix Products, the Uniqueness of the Inverse, and Block Matrix Products. Applications (MATLAB).



7. Coordinate Vectors, Basis Change.

Lecture Seven: Matrix Representations with Respect to General Bases. Theory: Rank, Matrix Transpose. Applications: Subspace Basis Change, Calculus.



8. Determinants, Lambda-Matrices.

Lecture Eight: Laplace Expansion, Gaussian Elimination, and Properties. Theory: Axiomatic Definition. Applications: Volume Wronskian.



9. Matrix Eigenvalues and Eigenvectors.

Lecture Nine, Using Vector Iteration: Vanishing and Minimal Polynomial, Matrix Eigenanalysis, and Diagonalizable Matrices. Lecture Nine, Using Determinants: Characteristic Polynomial, Matrix Eigenanalysis, and Diagonalizable Matrices. Theory: Geometry, Vector Iteration, and Eigenvalue Functions. Applications: Stochastic Matrices, Systems of Linear DE's and MATLAB.



10. Orthogonal Bases and Orthogonal Matrices.

Lecture Ten: Length, Orthogonality, and Orthonormal Bases. Theory: Matrix Generation, Rank 1 and Householder Matrices. Applications: QR Decomposition, MATLAB, and Least Squares.



11. Symmetric and Normal Matrix Eigenvalues.

Lecture Eleven: Matrix Representations with respect to One Orthonormal Basis. Theory: Normal Matrices. Applications: Polar Decomposition, Volume, ODEs, and Quadrics.



12. Singular Values.

Lecture Twelve: Matrix Representations w.r.t. Two Orthonormal Bases. Theory: Matrix Approximation, Least Squares. Applications: Geometry, Data Compression, Least Squares, and MATLAB.



13. Basic Numerical Linear Algebra Techniques.

Lecture Thirteen: Computer Arithmetic, Stability, and the QR Algorithm.



*14. Nondiagonalizable Matrices, the Jordan Normal Form.

Lecture Fourteen: (Jordan Normal Form). Theory: Real Jordan Normal Form, Companion Matrix. Applications: Linear Differential Equations, Positive Matrices.



Epilogue.


Appendix A (Complex Numbers and Vectors).


Appendix B (Finding Integer Roots of Integer Polynomials).


Appendix C (Abstract Vector Spaces).


*Appendix D (Inner Product Spaces).


Solutions.


Index.


List of Photographs.

These online resources are available at no cost.

Companion Website - Uhlig

http://www.prenhall.com/uhlig

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Transform Linear Algebra

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$119.99 $113.99 | ISBN-13: 978-0-13-041535-6

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