Let u = (1 , 0, 2, 1).
Find a basis for the subspace of vectors orthogonal to u in R4.
Please help me get started. The ones I did before used the Gram- Schmidt process, however I only have one vector, so I am unsure of how to begin.
Thanks.
Those vectorswhich are orthogonal to
satisfy the equation
. Now you can let, say,
,
,
and then we have
. Thus your subspace consists of all vectors of the form
. It's a 3-dimensional vector space (3 parameters) so you'll need 3 vectors in your basis; by picking values of
it's easy to find 3 vectors who do the job. Take it from there!
Ok, I think I got it now.
Example, if I were to let u = 1 and v = w= 0. then x4 would be -1 and that would be one of the vectors in the basis.
So a solution would be
( 1 0 0 -1)
(0 1 0 0)
(0 0 1 -2), am I right? I am unsure if the second one is correct, but I think it makes sense to me.
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