Guide for Week 3
Math 408, January 20, 2020
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Reading Assignment:
Homework Assignment:
Vocabulary Words
- Linear Least Squares Problems
- the linear least square problem
- polynomial interpolation by linear least squares
- the normal equations
- Show Null(A)=Null(A^TA)
- 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
- how to solve the the least norm solution problem to $Ax=b$ using the QR factorization
- Optimization of Quadratic functions
- quadratic functions
- the relationship between linear least squares and quadratic functions
- symmetric matrices
- unitary matrices
- diagonal matrices
- positive/negative definite/semi-definite matrics
- the existence and uniqueness theorem for quadratic optimization
unconstrained case
- the existence and uniqueness theorem for quadratic optimization
with affine constraints
- Choleski factorization
- 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
- positive definite/semi-definite matrices
- the existence and uniqueness theorems for quadratic optimization
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Skills to Master:
- Forming and solving the normal equations
- Computing orthogonal projections
- Computing the QR factorization
- using the QR factorization to solve the normal equations and obtaining least norm solutions
- solving a quadratic optimization problem
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Quiz:
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The quiz will consist of 2 questions.
The first question will be related to the vocabulary
words for the linear least squares problem
with an emphasis on the Gram-Schmidt, the QR-factorization
and how they are used to solve the linear least squares problem.
This material is contained in Chapters 2 of the course
Notes.
You are responsible for all of Chapter 2 except for section
5.3 on the use of Householder reflections to compute the full
QR-factorization.
The second question will be computational
and will be similar to the problems in
Problem set 2.