Question

linear algebra

(3) Find the QR factorization of A.

Thank you for the help.

Answer 1

[ 1 3 A = - NN 4 4 Let Ag = [ 2, 2, 1]', Q2 = [ 3, 4, 1]. Now, perform the Gram-Schmidt procedure Ug = Q4 -. [ 2, 2, 1]" est House 5 € = { [ 2, 2, 1) U₂ = 0₂ -(0.eg) e

10 4 Uz = [3, 4, 1]' - 16+8+1).} } [2, 2, 1]" - [ 3, 4, 1]'- $ [ 2; 2, 1]" = (3-40, 4-10, 1-3] ; U = [- £, £, - }] = { [-3, 21 -2]" Beethod Prefp=12943)}$(-1, 2, -21" = f(-1, 2, -237 L > U2 = .. . Thus, D= [ ] Q = NN 2 and, e R = r ages agen) To.azi @s .

Qg. @s = { [ 22+2° +12) = 3 az: es = {( 3x2 + 4x2 +1x 1) = f(6+8 + 1) = 5 az.bz = { [3X(-1) + 4 x2 +1 X(-2)] = { 1-3 +8-2) = 1 50, Buſ 3 5 7 Hence, the OR factorization of matrix A is [ 2 31 1 2 4 1 | 1 1]. 1 2 -17 1 2 2 1 -2] To solve the matrix equation T 2 3 7 rail 1 2 4 Il A x = b A = QR ORX= b RX = 0-4b-016 since Q is an orthogonal matriz.

2 2 1 50. Rx = 1 1 1 2X1 + 2X2 + 1 X 1 حالي -1X1 + 2X2 - 2X1 ا 77 2+4+1 حولي) 1 -1 +4-2 RX = WA or 3 5 7 127 10 1] [22] حالي) Use backward substitution I2 = and 37, +522 = } → 324= 3 I = 3 HIM voor