Orthogonalization (Gram--Schmidt)

* Finding the closest point in a plane

* Closest-point problem in in higher dimensions

** Closest point and coefficients

** Closest point and least squares

* High-dimensional projection onto/orthogonal to

** High-Dimensional Fire Engine Lemma

** Finding the projection of b orthogonal to Span {a1, ..., an}

** project orthogonal(b, vlist)

*** project orthogonal(b, vlist) doesn’t work

*** How to repair project orthogonal(b, vlist)?

*** Don’t fix project orthogonal(b, vlist). Fix the spec.

*** Loop invariant of project orthogonal(b, vlist)

*** Proof of loop invariant of project orthogonal(b, [v1 , . . . , vn ] )

*** Correctness of project orthogonal(b, vlist)

** Augmenting project orthogonal

* Building an orthogonal set of generators

** The orthogonalize procedure

** Correctness of the orthogonalize procedure

* Matrix form for orthogonalize

** Example of matrix form for orthogonalize

** Solving closest point in the span of many vectors

* Mutually orthogonal nonzero vectors are linearly independent

* Computing a basis

* Orthogonal complement

** Computing the orthogonal complement

* Augmenting orthogonalize(vlist)

* QR factorization

** Matrices with mutually orthogonal columns

** Using the QR factorization to solve a matrix equation Ax = b

* The least squares problem

** Representation of b|| in terms of columns of Q

** QR Solver Algorithm for Ax\approx b

** Application of least squares: linear regression

*** Linear regression

*** Application of least squares: coping with approximate data

*** Application of least squares: industrial espionage problem

*** Application of least squares: Sensor node problem

*** Applications of least squares: breast cancer machine-learning problem

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