Question

1. (4 marks) Show that the set of vectors {ős = (1,1,1,1), in = (1,0, -1,0), öz = (0,1,0, -1)} is orthogonal. Use those vectors in the set to get an orthonormal set {wi, wz, ws}.

Answer 1

Pove... In Here inner product simply is a dot produs Sue know that the set of Vectors, let s= are said to be to be orthogonal if <V , &> =0 for itj Here <u, v <u, v; - represent the Inner product of Vit Vj Since we are given that ( t, d, 1, 1) =(1,0,-1,0) (0,1,0,-1) WEV - then <v, u u > - <(2, 4, +, I) . (2,0-2,0) tto-1 to = 1-1 le C 27,,? > = 0

since we know that LU , Ves = 55, is an R Hence <u> a < > 20 so it is clear that de are orthogonal vectors similary <s, va > <(1,0,-1,0), (0,1,0,.237 o tototo <%, 37= Hence & Ve are also orthogonal <V ₂ % ₂ >=0 vectors. Now we will chech at V 2 <ig, v> ={(6,1,0,4),(2,1,1,1)> = 0 + 1 +0 -1 1 - 1 Luz ü>

V are also so *Hence & orthogonal the set { i, u, uz}, is the set of orthogonal rechers ů Hence proved I) Now we lectors ie, ie, iso will set the orthonormal by using 14, ull 90 Sincee know that the lectors To wa... con said to orthonormal if <Wi, wi> if izj 1 cit since our given set { e il is orthogonal, so if normalize 1 3

each vectors, then ule of its we get orthonomal vectors is, , i z. for normalizing I u s Just multiply by the reciproccel length Clength). will find the length of ", is a rectors get ll 2,11 = vo ve Now we then 3. = 1+1+1+1 2 80 length of ₂ = 2

Similarly Il la 11 v <ius - Jit 1to+1 to ~ Illall V2 length of ve again Il üzl - vo o+1+0+1 - VŽ length of V3 Hence w 그 ď 1 2. 1/2 (+ , 2, 1) = ( 2 , 3 , 2 , 3 ) B = (1,1,5,6)

similarly ull get Iů 2 Ja = F (1,0,-1,0) = (-1/2 ,0 , 7 / 0) = Ageein - ] LO3 VE = I (0,1,0,-1) V2 느 an Flor Ho 7/2) thence 2 ,= (4,4,41), 4(4,0) 23= (0, 1/3 , 0, 7 / 2 is prinshormel set of io, 2
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