Question

0 2. (10 points) Find a basis for the orthogonal complement of span in RS

Answer 1

Let set S= 2 o 1 -2 4 and we span(s) is a subspace of Rs. - 2 2 1 -1 We have to find a basis for the orthogonal complement of this W = span (5) Note that s is ca diseally dependent set.

Let de can element of basis for the Aethogonal "complement of w=span ) a b and so to the sectors of set s is orthogonal their dost products with will be o. e jie 1 2 of i a bo C OJNY 0 - 2 -1 2 d e >> 9-25+06+20 te=0 ; 2a +06-26-dte=0; Da +46-20-50-e=0 1 2 2 a O > (I) 2 O 1 b -1 -5 с -1 0 -2 4 e Solve this system using Gaussian Elimination Method: R3 R3-R2 1-2 02 1 04 -2 5 -1 0 2 1 4-2-5-1 4 -2 -5 -1 I-2 2 1 2 0 -2 -1 1 4 -2 -5 -1 1 -2 0 1 Ray 7 R2 ₂ R1+2 R2 aglo -1 / 42 1 -42 -574 0 1 4/4 0 0 0 0 0 0 0

System is equivalent to a 오 외2 12 0 1 예2 a b C d 0 NIP e =0 t a-C-| - - 어 - e =o "a & & are leading Kariables and c, d, e are free variables Thus we have: a C 아들 - 42 1 7외2 5대 외12 e 들다름이 C =C 외 1 0 Anto C 0 4 e 0 e 1 - Basis of orthogonal complement to Span(s) is: 1 외2 -12 4 1 외니 0 4 0 1