ekiim's blog

# Linear Algebra

## Content

• Preface
1. Matrices and Gaussian Elimination
1. Introduction
2. The Geometry of Linear Equations
3. An Example of Gaussian Elimination
4. Matrix Notation and Matrix Multiplication
5. Triangular Factors and Row Exchanges
6. Inverse and Transpose
7. Special Matrices and Applications
• Review Exercises
2. Vector Spaces and Linear Equations
1. Vector Spaces and Subspaces
2. The Solution of $m$ Equations in $n$ Unkowns
3. Linear Indepencance, basis, and Dimension
4. The four Foundamental Subspaces
5. Networks and Incidence Matrices
6. Linear Transformations
• Review Exercises
3. Orthogonality
1. Perpendicular Vectors and Orthogonal Subspaces
2. Inner Products and Projections Onto Lines
3. Projections and Least Squares Approximations
4. Orthogonal Bases, Orthogonal Matrices, and Gram-Schmidt Orthogonalization
5. The Fast Fourier Transform
6. Rreview and Preview
• Review Exercises
4. Determinants
1. Introduction
2. The Properties of the Determinant
3. Formulas for the Determinant
4. Applications of Determinants
• Review Exercises
5. Eigenvalues and eigenvectors
1. Introduction
1. The Diagonal form of a matrix
2. Difference Equations and the Powers $A^k$
3. Differential Equations and the Exponential $e^{At}$
4. Complex Matrices: Symetric vs Hermitian and Orthogonal vs Unitary
5. Similarity Transformations
• Review Exercises
6. Positive Definite Matrices
1. Minima, Maxima, and Saddle Points
2. Test for POsitive Definiteness
3. Semidefinite and Indefinite Matrices; $Ax = \lambda Mx$
4. Minimum Princples and Rayleight Quotient
5. The Finit Element Method
• Review Exercises
7. Computations with Matrices
1. Introduction
2. The Norm and condition number of a matrix
3. The Computation of Eigenvalues
4. Iterative Methods for $Ax = b$
8. Linear Programming and Game Theory
1. Linear Inequalities
2. The Simplex Method and Karmarkar’s Method
3. The Theory of Duality
4. Network Models
5. Game Theory about the Minimax Theorem

Appendix A) The Singular Value Decomposition and the Pseudoinverse

Appendix B) The Jordan Form

Appendix C) Computer Code for Linear Algebra