This is a question from Pinter's A Book of Abstract Algebra p 289
He asks for a basis of the subspace of $\mathbb{R}^3$ spanned by the set of vectors $(x,y,z)$ that satisfy $x^2+y^2+z^2=1$
I don't really understand this questions but it seems like since the sphere is a surface it should be a basis with two elements that are linearly independent.
So can we just choose $(1,0,0),(0,1,0)$ as our basis? They are in the space and linearly independent but how would we ever write say $(\sqrt{(1/3)},\sqrt{(1/3)},\sqrt{(1/3)})$.
Edit: This is exercise C.6 of Chapter 28 (Vector Spaces) of Pinter's book. The exact wording of the exercise is:
Find a basis for the subspace of $\mathbb R^3$ spanned by the set of vectors $(x,y,z)$ such that $x^2 + y^2 + z^2 = 1$.