# Find Basis orthogonal to a given u

• Apr 4th 2010, 10:10 PM
msfynestrini
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.
• Apr 4th 2010, 11:29 PM
Bruno J.
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!
• Apr 5th 2010, 12:55 AM
msfynestrini
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.
• Apr 5th 2010, 09:42 AM
Bruno J.
That's good! (Nod)
• Apr 5th 2010, 04:16 PM
msfynestrini
Thank you soo much