Thread: Find Basis orthogonal to a given u

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 vectors $\displaystyle (x_1, x_2, x_3, x_4)$ which are orthogonal to $\displaystyle (1,0,2,1)$ satisfy the equation $\displaystyle x_1+2x_3+x_4=0$. Now you can let, say, $\displaystyle x_1=u$, $\displaystyle x_2=v$, $\displaystyle x_3=w$ and then we have $\displaystyle x_4=-u-2w$. Thus your subspace consists of all vectors of the form $\displaystyle (u, v, w, -u-2w)$. It's a 3-dimensional vector space (3 parameters) so you'll need 3 vectors in your basis; by picking values of $\displaystyle u,v,w$ 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.

That's good!

Thank you soo much