MATH 415 Study Guide - Final Guide: Alfred Von Wurzbach, Null Character, Design Matrix
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Suggested practice exercises: chapter 3. 3: 3, 5, 6, 13, 24, 25. Strang lecture: lecture 16: projection matrices and least squares. Review: suppose that w is a subspace of rn with orthonormal basis u1, . , up, and x is a vector in rn. = {x1a1 + x2a2 + + xxan | x rn} = {ax | x rn} X is a least squares solution of the system ax = b if x is such that a x b is as small as possible. Ax = b is consistent b is in col(a) so if ax = b is inconsistent we: replace b with its projection bw onto w = col(a), and solve a x = bw . (consistent by construction!) Otherwise, we could not proceed in the same way. Hence the projection of bw of b onto w = col(a) is bw = We have already solved a x = bw in the process: x = (cid:20)1/2.