Question

Give the least squares solution to the over-determined system of linear equations: 2 = 5 +02 = 1 2 2 :r2 = 2 2 22 = 3 In what sense is this better than the exact solution 2 = 5, 02 = -4 to the first two equations?

Answer 1

Friday Given system of linear eques is a dit ora 25 - 2, + % 1 - 22-29, ? — from t ; & as 2 2 - 4 but (2, 2) = (5-4) doesn't satisfy equos PO. - Hence, synem is not solvable. So, we will find lealt square soluhon. 3 & Writing system of formy cour ① to W in matrie when = AX a be AXC b is not solvable TAX a Ab in least we square solve method.

1, -2 od -2 -1 6716: [0 => 421 -2 =5 l on solvity, we get 1-281 +6%2=0 t 2 a 312 Anes ng ? lealt square If we foon consider 7 egun to egun as ft. Irres, then exact soln (5, 4) die lie on 1st two lines. So, its distance from a bit too lines is 7ero. but (Xigh ) n (5-4) in line ④ & 7 5-21-4)- ll (line © 5+4 + 3 a 12 Clime ♡ '

. - Shortent line diatance from à point il always borpendicular to a straight cintance a a I It perpendicular distance of dine ax, a blat c a Taxi + bx + c point (2,dly) to o is I points of lour 2,05 T lines X 4X=1 O • 1972 T Distance I from point 5 ,-4) (3 of equal Shorkest distance is encircled a Clearly. (314) is at lesser dixtame from most of the lines then that of 15,14),