## 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.

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