Scientific Computing

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CS205 Homework #2 Problem 1 [Heath 3.29, page 152] Let v be a nonzero n -vector. The hyperplane normal to v is the (n-1)-dimensional subspace of all vectors z such that v T z = 0 . A reflector is a linear transformation R such that Rx = - x if x is a scalar multiple of v , and Rx = x if v T x = 0 . Thus, the hyperplane acts as a mirror: for any vector, its component within the hyperplane is invariant, whereas its component orthogonal to the hyperplane is reversed. 1. Show that R = 2P - I , where P is the orthogonal projector onto the hyperplane normal to v . Draw a picture to illustrate this result 2. Show that R is symmetric and orthogonal 3. Show that the Householder transformation H = I - 2 vv T v T v , is a reflector 4. Show that for any two vectors s and t such that s = t and s 2 = t 2 , there is a reflector R such that Rs = t Problem 2 Let A be a rectangular m × n matrix with full column rank and m > n . Consider the QR decomposition of A . 1. Show that P 0 = I - QQ T is the projection matrix onto the nullspace of A T 2. Show that for every x we have Ax - b
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