Parts C-I

Approximationg t' is silly, but will make for easy integrals. -2 4 13 0 -1 2 3. Consider A = 0 3 47 and b = 13 -2 21 25 (a) (2pts) Compute AT A. (Use a machine if you like, but record the answer here.) (b) (2pts) Compute AT b. (Use a machine if you like, but record the answer here.) (c) (6pts) Using (a) and (b) find all least squares solutions to Ar = b. (You may use a machine for the row reduction, but be clear to show what matrix is being row reduced.) If there are infinitely many solutions, express them in parametric vector form. (d) (4pts) Using part (c), find the projection of b onto the column space of A, that is, b = projcol() b. Tip: Recall that a least squares solution is a vector r such that A x is as close to b as possible. (e) (4pts) Compute the least squares error. 1001 0101 (f) (2pts) Based on the fact that the RREF for A is 0 0 1 1 give a basis for Col(A). Be sure to use { } to list the vectors in the basis. 0 0 0 0 (g) (6pts) Use Gram-Schmidt to convert the basis found in part (f) to an orthogonal basis for Col(A). (h) (4pts) Make use of the orthogonal basis you found in part (g) to find the projection of b onto the column space of A, that is, b = projcol(A) b. You should, of course, get the same answer you did for part (d), so make sure you clearly demonstrate the use of the orthogonal basis. (i) (4pts) Using part (h) find all least squares solutions to Ax = b. Be sure to use part (h) and not just recopy your work from part (c). (You may use a machine for the row reduction if you like, but be clear to show what matrix is being row reduced.) If there are infinitely many solutions, express them in parametric vector form