Guide for Week 2/3
Math 408 Section A, Jan. 15, 2016
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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
- 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
- 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, one conceptional and one computational.