MATH 131 Midterm: MATH 131 UMass Amherst Umass_Exam2_2017_Sol
Document Summary
Spring 2005: let a be an orthogonal n n matrix (recall this means that the columns of a are orthonor- mal), and let a be its transpose. (a) find: Aa = in [this is equivalent to columns of a being orthonormal]. Yes, since the rows of a are the columns of a , which we just saw is orthogonal. (e) the qr-factorization of a is given by: Q = a [since a is already orthogonal, so its columns make up q]. R = in [since a = qr gives r = q 1a = a 1a = in]: find all orthogonal matrices of the form. The orthonormality conditions ~v1 ~v3 = 0, ~v2 ~v3 = 0, ~v3 ~v3 = 1 yield: 2 c = 0 a + b c = 0 a2 + b2 + c2 = 1. The rst two equations give b = c and a = c b = 0.
