Question

Solve each of the following linear systems by the method of the example on page 283.

(a) 3x − y = 4        x + 5y = 9

(b) x − 2y + z = 1       y + z = 0         −x + y + z =−3

HInt; first find the QR decomposition of the coefficient matrix via the Gram–Schmidt process

Answer 1

(a)Given 3x-y-4,x+5y 9 The matrix form of the system is AX =b. here 4 = Here a = | , 1.4 = Performing the Gram-Schmidt procedure , we get 4 9 1 (3 3/V1O -1 -13/103/10 -8/5 1 8/5 10 u264 576 24/5) 16 24/5)3 25 25 10 102 1 -1) до 3V1010 10 То To 10 10 -11016 8 a,e, 10 аге, 10 Finally A-QR1 30 To 10 10

Now compute Y Qb,here Y-1 O O1010 10 V10 (4 10 23 10 10 /10 now solve for RXY 21 10 23 10 10 23 10 23 16 on backward substitution hoy- y- 21 Now plug y in v10x+-V10y- 21 1 10 5 21 1 23)29 16 16 10 29 16

(b)Givenx -2y+z-1,ytz-0,-xty+z-3 The matrix form of the system isAX-b, 1 -2 1 Performing the Gram-Schmidt procedure , we get 1+0+1 141 2 1/2 1/2

1, +1+1-1/2 4 4 -リ-LE 1,6 1,6 1,6

Now compute Y-erb, here Y =| ½ now solve for RX-Y 2v2 on backward substitution /5z -25»1--2 Now plug y,z in V2x--v2y-242 23