[Apostol's Calculus] Exercises 1.5: Solution to 25

Let S be the set of all vectors v such that vv

Proof:

Assume S is a real linear space, then S must be closed under addition (axiom 1). In other words, if v and w are in S, v+w must also be in S. By that logic, if v is in S, v + v = 2v must also be in S. Another way of saying this is that S must be closed under scalar multiplication (axiom 2).

But for all v in S, v

Yet multiplying v by any constant c, breaks that form. So S cannot possibly be closed under scalar multiplication or addition.

Therefore, S cannot be a real linear space.

QED