Let U be the set of all vectors in R3 that are perpendicular to a vector u. Prove that U is a vector space.
I have no idea where to even start. :S
It is enough to show that U is a subspace.
You need to check that if a,b in U and ka in U for every real number k
(i.e., need to check if you have two vectors that perpendicular to u, then the sum of these vectors still perpendicular to u and if you have a vector that perpendicular to u then the scalar multiple of the vector still perpendicular to u)
the problem is, what kind of creatures do we check? well, two vectors are perpendicular if their dot product is zero. so that gives you a way to characterize the vectors in U, and now we know what to perform our checklist with