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2 the20vector b random structures least squares error

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Unformatted text preview: 12 25 IX.4 presentations) Continuous Limit Laws Marni Remark. What is b − Av for a given Sophie vector v ∈ Rn ? Quasi-Powers and 13 30 IX.5 Gaussian limit laws Asst #3 Due If Ax = b was consistent, and v is a solution, then b − Av = 0. 14 Dec 10 Presentations So b − Av is a vector that represents how different from a solution to this system v is: if b − Av = (e1 , . . . , en ), then ei is a scalar that represents how the ith component of Av is far from the ith component of b, which is why it is called the error vector. ˆ This is why we try to find a vector x that minimizes the length of this error vector, which is exactly e1 + · · · + e2 . n Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY 2 Version of: 11-Dec-09 Theorem (7.8.3(a,b)). (a) The least-squares solutions of a linear system Ax = b are exactly the solutions of the linear system AT Ax = AT b. If A has full column rank, the normal equation has a unique solution given ˆ by x = (AT A)−1 AT b. Ter...
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