Question

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 ⊥.

Answer 1

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.

For further reading and examples, see for example
http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process