Question

3 5 1 -1 -3 1 1. Let A b 1C Col(A) 0 2 1 5 2 1 5 8 a) Find an orthonormal basis for C b) Find a QR factorization for A. c) Find b projc(b). Does b= b? What does this tell us about the number of solutions to Ax = b?

Answer 1

-1 -3 1 1(a) Given A= 0 2 3,6= 1,C = col(A) 1 5 2 1 5 8) Orthonormal basis for col(A): First we need to find col(4): (135) (1 3 5 (135) 1-1 -3 1 R+R OO6RERO 2 3 R (1/2) Rref A = 0 2 3 0 2 3 006 1 5 2 R-R, O 2 -3 O2 -3 (1 5 8R-RO 23) 023) (1 3 5 R-3R (1 O 1/2) (1 0 1/2 R-(1/2) (1 00 0 1 3/2 0 1 3/2 0 1 3/2 R, -(3/2) 0 1 0 00 6 00 6 R (1/6) 0 0 1 0 0 1 0 2 -3 R.-2R, 0 0 -6 OO -6 R4 +6R000 0 2 3 R: -2R, 0 0 0 000 1000) From tref (A) first second and third columm have a pivot element 1 (1 ) (3 ) (5) Therefore the col(A)= 0 in un

Let u= 0 = imman ,w= step 1: produces an orthogonal basis: =n=D 0 buva) (aa) -TO- -TO- - TO 1 / 1 (wa), (wb) Ta-ala 70.6) CW - TO ---TO- -TO- را برا ہ OL100 رلا دیا

step 2.produces an orth onormal basis by replacing each vector with avector of norm 1: replace a with a 1+1+0+1+1 replace b with V1+1+4+1+1 NING 6 G G5 replace c with درا را ہ را نیا 19+9+0+9+9 HISO Š flm S Therefore the orthonoramal basis for C = 10 16-1010-1010 G -1010 v

(1 3 5 -1 -3 1 (b)Given A = 0 2 3 1 5 2 1 5 8) nama -1 -3 let v = 0 ,2 = 2 and v2 = 3 IT From(a) we have the orthonoramal basis = { OOOO Ni-NIM 50,Q=(4 92 93) = O IVI NI NIN O NIENIM 19 vgv.gi (2 8 7) R= 0 v,92 v:q2 = 0 2-2 312 (O 09:v) 0 0 6 A=QR = MINILO NANIN (28 ) lo 212 312 lo 0 6 Nivin