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 ⊥.

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.