Guide for Week 5
Math 408
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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
- Optimization of Quadratic functions
- quadratic functions
- the relationship between linear least squares and quadratic functions
- symmetric and self-adjoint matrices
- unitary matrices
- diagonal matrices
- positive/negative definite/semi-definite matrics
- Lemma 3.1 page 29 of the course notes.
- the existence and uniqueness theorem for quadratic optimization (unconstrained)
- the existence and uniqueness theorem for quadratic optimization on an affine set
- the existence and uniqueness theorem for quadratic optimization with linear constraints
- the optimal value in a quadratic optimizations problem when an optimal solution
does not exist
- Choleski factorization
- the generalized Choleski factorization
- H-conjugate vectors
- the conjugate directon algorithm and the Expanding Subspace Theorem
- 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 theorem for quadratic optimization
- the existence and uniqueness theorem for quadratic optimization on an affine set
- the existence and uniqueness theorem for quadratic optimization with linear constraints
- the conjugate gradient algorithm
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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
- solving a quadratic optimization problem
- computing a Cholesky factorization
- implementing the conjugate gradient algorithm
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Midterm: