And in general how can I show wether a subset of a vector space is affine or not?
Whatever the overall space is. In your problem, it would be . As long as you are working in you can charactize an "affine set" (that is not a subspace), geometrically, as a line or plane or "hyper-plane" (of whatever dimension) that does NOT include the origin. The condition that simply says that this is the set of all vectors in with length . Is that a plane (of whatever dimension) in ?
Hmm, yes I tihnk that is a plane. By the way I found an other way of solution but Im not sure wether it is right or not. So is affine if for all and .
And what I did:
and because and
So it is an affine set.