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Let W be the subspace of R5 spanned by the vectors w1, w2, w3, w4, w5, where

w1 = 2 −1 1 2 0 , w2 = 1 2 0 1 −2 , w3 = 4 3 1 4 −4 , w4 = 3 1 2 −1 1 , w5 = 2 −1 2 −2 3 . Find a basis for W ⊥.

Verified Answer :

syed514

To find W⊥, you can use the Gram-Schmidt process using the usual inner-product and the given 5 independent set of vectors. 

Define projection of v on u as 
p(u,v)=u*(u.v)/(u.u) 
we need to proceed and determine u1...u5 as: 
u1=w1 
u2=w2-p(u1,w2) 
u3=w3-p(u1,w3)-p(u2,w3) 
u4=w4-p(u1,w4)-p(u2,w4)-p(u3,w4) 
u5=w5-p(u4,w5)-p(u2,w5)-p(u3,w5)-p(u4,w5) 

so that u1...u5 will be the new basis of an orthogonal set of inner space. 

However, the given set of vectors is not independent, since 
w1+w2=w3, 
therefore an orthogonal basis cannot be found. 

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