MATH115 Study Guide - Quiz Guide: Orthogonal Matrix, Orthogonal Complement, Eigenvalues And Eigenvectors

• School: University of Waterloo
• Department: Mathematics
• Course: MATH115
• Professor: Martin Pei

Document Summary

Is not in s. determine the vector (cid:126)s in s that is closest to (cid:126)x, that is, nd (cid:126)s that. 2 = w2 projv1 (w2) = v(cid:48) 3 = w3 projv1 (w3) projv2 (w3) = v(cid:48) 9 v3 = (b) here we just nd the projection of (cid:126)x onto s. where the coe cients a, b and c are: projs ((cid:126)x) = a (cid:126)v1 + b (cid:126)v2 + c (cid:126)v3. 18 a = (cid:126)s = a (cid:126)v1 + b (cid:126)v2 + c (cid:126)v3 = c = 0. 2/3 1/3 (c) the dimension of s is 1, so we just need one vector orthogonal to s. the perpendicular component of (cid:126)x relative to s will su ce. Scaling by 3: {7 marks} orthogonally diagonalize a = (cid:126)x (cid:126)s = We rst nd the eigenvalues of a. det(a i) = 6: {4 marks} suppose you apply the gram-schmidt procedure on the vectors { (cid:126)w1, .