Question

full proofs for both and please write legibly

5. Let T be an orthogonal transformation on a finite dimensional vector space V over the real numbers, with an inner product. Show that D(T) = $1. 6. Show that if u,...,U, are orthonormal vectors in R, (see (15.7)), then D(uj, ..., Un) = 1.

Answer 1

5 Since I is orthogonal transformation then associate matrix of I say A is orthogonal. Then DCT) = D(A). Now since A is orthogonal, AAT = 1 = ATA by definition using the fact that det (AB) = det (Aldet (B), we have det(1) -1 = der (AAT) = det (A) det(AT) r det (AT) adeta] of det (A) det(AT) = 1 or det (A) det(A) = 1 or det (A)] = 1 or det (A) at 1 Therefore D(A) = Il so D(T) = 1)

(6) We know a result that a matrix A is orthogonal if and only if the row vectors of A form an ortho normal basis of rn under the cuclidean inner product and A is orthogonal iff At is orthogonal. Here A = [1,..., Un], AT = Ses at 497 AA - E AT (ATT = AT A = ...un SU Je, u ... 4, Un I und ... Man uni

at follows that ATA = I iff i. U as if it ui. Ug = Yo if it's ( Now since UI, U,... Up ince UI, U,... Un are ortho normal. Hence ATA=I AT is orthogonal A is orthogonal Hence by (5) det (A) = 11 D(C,;..., g = 11