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)
start with verifying that U has all the properties of a vector space. this is just going through a check list.
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