Guide for Week 4/5
Math 408 Section A
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Reading Assignment:
Finish reading Chapter 3 of the book and most of Chapter 4.
Homework Assignment:
Vocabulary Words
- Linear Least Squares Problems
- the linear least square problem
- polynomial interpolation by linear least squares
- the normal equations
- uniqueness in the normal equations
- orthogonal projections
- QR factorization
- orthogonal projections onto the 4 fundamental subspaces
- how to solve the normal equations using the QR factorization
- Optimization of Quadratic functions
- quadratic functions
- the relationship between linear least squares and quadratic functions
- symmetric matrices
- orthogonal matrices
- diagonal matrices
- eigenvalue decomposition for symmetric matrics
- positive/negative definite/semi-definite matrices
- the Rayleigh-Ritz Theorem
- the existence and uniqueness theorem for quadratic optimization
- the conjugate gradient algorithm
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Key Concepts:
- Linear Least Squares Problems
- the normal equations
- orthogonal projections
- the QR factorization
- Optimization of Quadratic functions
- the relationship between linear least squares and quadratic functions
- eigenvalue decomposition for symmetric matrics
- the Rayleigh-Ritz Theorem
- positive definite/semi-definite matrices
- the existence and uniqueness theorem for quadratic optimization
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Skills to Master:
- Forming and solving the normal equations
- Computing orthogonal projections
- using the QR factorization to solve the normal equations
- computing the eigenvalue decomposition for a symmetric matrix
- solving a quadratic optimization problem
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Quiz:
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The quiz will consist of 2 questions related to the content (covered in class) of Chapter 4 of the course notes.