[Apostol's Calculus] Exercises 1.5: Solution to 23

Let S be the set of all vectors v = <x, y, z> in V₃ with either x = 0 or y = 0. Assume S is a real linear space. Then, S must be closed under addition (axiom 1). Let v₁ = <0, b, c> ∈ S and let v₂ = <a, 0, c> ∈ S, where a, b, and c are non-zero. Since S is a real linear space, v₃ = v₁+v₂ must be in S. But v₃ = <a, b, 2c>, which has neither a zero x-component nor a zero y-component, since a, b, and c are non-zero. So v₃ cannot be in S. Thus, S is not closed under addition. Therefore, S cannot possibly be a real linear space.

QED