Guide for Week 2
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Reading Assignment:
Homework Assignment:
Vocabulary Words
- Math 308 Review:
- m by n systems of linear equations
- Gaussian elimination
- echelon form
- dot product
- orthogonal vectors
- subspace
- basis and dimension
- the 4 fundamental subspaces associated with a matrix
- rank and nullity
- properties of matrix multiplication
- manipulation of block structured matrices
- Gaussian elimination matrices
- the Fundamental Theorem of the Alternative
- 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
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Key Concepts:
- Math 308 Review:
- Linear systems of equations
- Gaussian elimination as a method to attain echelon form
- Echelon form and the solution of a linear system
- Subspaces and their representations: internal and external
- Properties of the 4 fundamental subspaces associated with a matrix.
- Gaussian elimination as matrix multiplication,
and the LU factorization.
- Manipulation of block structured matrices
- Linear Least Squares Problems
- the normal equations
- orthogonal projections
- the QR factorization
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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
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Quiz:
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The quiz will consist of 2 questions and will focus on the linear least squares problem LLS.
Specific topics to focus on are the formulation of LLS, the relationship to the normal equations,
as well as the existence and uniqueness theorem for LLS. In addition, you need to be prepared to use
facts about orthogonal projections and how to produce them for a given subspace.
The first question will be related to the vocabulary
words for the linear least squares problem.
This is Chapter 2 of the
Notes. We have yet to cover the QR factorization, so you are only responsible
for Sections 1, 2, and 3 in Chapter 2 as well as knowledge of the Gram-Schmidt
orthogonalization technique from Math 308.
The second question will be computational
focusing on the linear least squares problems, specifically,
%how to formulat polynomial fitting problmes as linear least squares problems,
the normal equations
and how to use them to solve linear least squares problems, and how to
compute orthogonal bases and orthogonal projections. You will not be asked to compute
a QR factorization. The kinds of problems you may be asked are similar to those appearing
in the second problem set excluding problems 2,3 4 parts (c) and (e).
All other homework questions are fair game. Solutions to this problem set can be found
here. Remember, that orthogonal bases can be obtained by applying the
Gram-Schmidt orthogonalization process you learned in Math 308.