if A is a nxn matrix and x, y, v are vectors in Rn

know that (Ax) ? (Av) = 0 and y ? v = 0

dose span{x}=span{y} ?

okmarks People Window Help 74% X y! crowdmark - Yahoo Search | x 7 Crowdmark X v/ Consider R2 and its subspac x + m/student/assessments/homework-5-aeb30 Q YW + Drag and drop your images or click to browse... Q5 (10 points) Let P be an orthogonal n X n matrix and L(x) = Px the associated linear map. (a) Show that if n = 2, then there is some 0 E [0, 2x] such that P is of one of the forms P = cos 0 - sin e sin 0 cos 0 or P = cos 0 sin 0 sin 0 - cos 0 (b) Show that any eigenvalue of P which is a real number is + 1 (don't just cite Theorem 9.2.7(4) here, but it is OK to use the other parts of Theorem 9.2.7). (c) Show that if n is odd, then there is a non-zero vector x E I" such that PX = x or PX = -x. (d) Show that if n = 3, then there is an orthonormal basis B of R and some 0 E [0, 2x] where the associated matrix [ ] , is of one of the forms +1 0 +1 0 [LIB = cos - sin e or [LB = cos e sin 6 sin 0 cos e sin 0 - cos 8 + Drag and drop your images or alick to browse. 101) STAT 231 - Fall 2019 Important Midterm A 9 MacBook Air0_LinAlgBook [o] B I 254 Chapter 9 Inner Products We now look at some useful properties of orthogonal matrices. THEOREM 7 If P and Q are n x n orthogonal matrices and X, J E R", then (1) (PX) . (PJ) = x. 3 (2) 11PXII = 11X'11 (3) det P = +1 (4) All real eigenvalues of P are 1 or -1 (5) PQ is an orthogonal matrix Proof: We will prove (1) and (2) and leave the others as exercises. We have ( PX) . ( Py) = ( PX) I ( Py ) = x p] Py = xy =x.y Thus, IPEIR = (PX) . (PX) = $.8=117112 0 Section 9.2 Problems [47 ([ 1 ] [2] [-2])