*2	Matrix Analysis (Basic Ideas)
*2.1	Basic Ideas from Linear Algebra
*2.1.1		Independence, Subspace, Basis, and Dimension
*2.1.2		Range, Null Space and Rank
*2.1.3		Matrix Inverse
*2.1.4		The Determinant
*2.2	Vector Norms
*2.2.1		Definitions
*2.2.2		Some Vector Norm Properties
*2.2.3		Absolute and Relative Error
 2.2.4		Convergence
*2.3	Matrix Norms
*2.3.1		Definitions
*2.3.2		Some Matrix Norm Properties
*2.3.3		The Matrix 2-Norm
 2.4	Finite Precision Matrix Comptutations
 2.4.1		The Floating Point Numbers
 2.4.2		A Model for FLoating Point Arithmetic
 2.4.3		Cancellation
 2.4.4		The Absolute Value Notation
 2.4.5		Round-off in Dot Products
 2.4.6		Alternative Ways to Quantify Roundoff Error
 2.4.8		Roundoff in Other Basic Matrix Computations
 2.4.9		Forward and Backward Error Analysis
*2.5	Orthogonality and the SVD
*2.5.1		Orthogonality
*2.5.2		Norms and Orthogonal Transformations
*2.5.3		The Singular Value Decompsition
*2.5.4		The Thin SVD
*2.5.5		Rank Deficiency and the SVD
 2.5.6		Unitary Matrices
*2.7	Sensitivity of Square Systems
*2.7.2		Condition
 
 3	General Linear Systems
 3.1	Triangular Systems
 3.1.1		Forward Substitution
 3.1.2		Back Substitution
 3.2	The LU Factorization
 3.2.1		Gauss Transformations
 3.2.2		Applying Gauss Transformations
 3.2.4		Upper Triangularizing
 3.2.5		The LU Factorization
 3.2.8		Solving a Linear System
 3.2.12		A Note on Failure
 
 Systems of Linear Equations: Error and Cost Analysis.
 3.3.1		Errors in LU Factors [just the main result]
 3.3.2		Triangular Solving with Inexact Triangles [just main theorem]
 3.4	Pivoting [just partial pivoting]
 3.4.1		Permutation Matrices
 3.4.2		Partial Pivoting: The Basic Idea
 3.4.4		Where is L? [PA=LU]
 3.5	Improving and Estimating Accuracy
 3.5.1		Residual Size Versus Accuracy
 
*4	Special Lineary Systems
 4.1.2		Symmetry and the LDL' Factorization [just form of factorization]
*4.2	Positive Definite Systems
*4.2.1		Positive Definiteness
*4.2.3		Symmetric Positive Definite Systems [esp. Cholesky]
 4.3	Banded Systems
 4.3.1		Band LU Factorization
 4.3.2		Band Triangular System SOlving
 4.3.3		Band Gaussian Elimination with Pivoting
 
 5	Orthogonalization and Least Squares
 5.1	Householder and Givens Matrices
 5.1.1		A 2-by-2 Preview
 5.1.2		Householder Reflections
 5.1.3		Computing the Householder Vector
 5.1.4		Applying Householder Matrices
 5.1.5		Roundoff Properties
 5.1.6		Factored Form Representation
 5.1.8		Givens Rotations
 5.1.9		Applying Givens Rotations
 5.1.10		Roundoff Properties
 5.1.11		Representing Products of Givens Rotations
 
 5.2	QR Factorization
 5.2.1		Householder QR
 5.2.6		Properties of the QR Factorization
 5.2.7		Classical Gram-Schmidt
 5.2.8		Modified Gram-Schmidt
 
 5.3	Full Rank LS Problem
 5.3.1		Implications of Full Rank
 5.3.2		The Method of Normal Equations
 5.3.3		LS Solution Via QR Factorization
 5.3.4		Breakdown in Near Rank-Deficient Case
 
 5.4	Other Orthogonal Factorizations
 5.4.1		Rank Deficiency: QR with Column Pivoting
 
*5.5	The Rank Deficient Least Squares Problem
*5.5.1		The Minimum Norm Solution
 5.5.2		Complete Orthogonal Factorization and x_{LS}
*5.5.3		The SVD and the LS Problem
 5.5.4		The Pseudo-Inverse
 5.5.6		QR with Column Pivoting and Basic Solutions
*5.5.8		Numerical Rank and the SVD
 
 
*7	The Unsymmetric Eigenvalue Problem
*7.1	Properties and Decompositions
*7.1.1		Eigenvalues and Invariant Subspaces
 7.1.2		Decoupling
*7.1.3		The Basic Unitary Decompositions
*7.2	Perturbation Theory
*7.2.1		Eigenvalue Sensitivity
*7.2.2		The Condition of a Simple Eigenvalue
 7.2.3		Sensitivity of Repeated Eigenvalues
*7.3	Power Iterations
*7.3.1		The Power Method
 7.3.2		Orthogonal Iteration [Subspace Iteration]
 7.3.3		The QR Iteration [algorithm sketch only]
 
 8	The Symmetric Eigenvalue Problem [and singular values].
 8.1	Properties and Decompositions
 8.1.1		Eigenvalues and Eigenvectors
 8.1.2		Eigenvalue Sensitivity [just how general thry reduces here]
*8.6	Computing the SVD [just the short intro section]
 
 10	Iterative Methods for Linear Systems
 10.1	The Standard Iterations
 10.1.1		The Jacobi and Gauss-Seidel Iterations
 10.1.2		Splittings and Convergence
 10.1.4		Successive Over-Relaxation