Transform Linear Algebra

Frank Uhlig. Born April 2, 1945, Mägdesprung/Harz; grew up in Mülheim/Ruhr, Germany; married, two sons. Mathematics student at University of Cologne, California Institute of Technology. Ph.D., CalTech, 1972; Assistant, University of Würzburg, RWTH Aachen, Germany, 1972-1982. Two Habilitations (Mathematics), University of Würzburg 1977, RWTH Aachen 1978. Visiting Professor, Oregon State University 1979/1980; Professor of Mathematics, Auburn University 1982. Two Fulbright Grants; (Co-)organizer of eight research conferences. Research Areas: linear algebra, matrix theory, numerical analysis, numerical algebra, geometry, Krein spaces, graph theory, mechanics, inverse problems. 40+ papers, 2+ books.

(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.