MATH 225 Spring 2017: Assignment 5 Due at 8:25 a.m. on Friday, June 16, 2017 It is important that you read the assignment submission instructions and suggestions available on LEARN. 1 0 1 1. (a) The set B = 0 , 0 , 1 is a basis for R3 . (You do not need to verify this.) Verify 1 1 0 that it is an orthogonal basis. (b) Without changing the direction of any of the vectors in B, change B into an orthonormal basis. (c) Let L : R3 R3 be the linear mapping is a rotation counterclockwise through the angle which 1 0. Use the orthonormal basis you found in part(b) to about the axis determined by ~ x = 4 1 find the matrix of L with respect to S, the standard basis for R3 . (You can take as given that the basis you have found is right-handed.) Page 1 1 0 0 2 0 0 2. Let W = Span 0 , 1 , 0. 0 0 1 5 3 2 0 5 (a) Determine if ~v = 6 is in W , the orthogonal complement of W. 3 2 (b) Find a basis for W . Page 2 3 4 1 0 , 4 , 1 . Use the Gram-Schmidt Procedure to find an orthonormal basis 3. Let S = Span 1 1 6 for S. Page 3 4. Verify that the following system A~x = ~b is inconsistent and then determine the vector ~x that minimizes ||A~x ~b||. x1 + x2 = 4 x1 + 2x2 = 1 2x1 x2 = 2 Page 4 5. Consider the data t y 2 1 0 1 . 9 3 1 3 (a) Find a and b to obtain the best-fitting equation of the form y = a + bt for the given data. (b) Find c and d to obtain the best-fitting equation of the form y = ct + dt3 for the given data. (c) Make a graph showing the data and the best-fitting curves from parts (a) and (b). Page 5