Which theorem could help prove one of these criteria from another?
a) If S = {u1, ... , up} and each uj has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
b) If S = {u1, ... , up} is an orthogonal set of nonzero vectors in Rⁿ, then S is linearly independent and hence is a basis for the subspace spanned by S.
c) If S = {u1, ... , up} and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
d) If S = {u1, ... , up} is a basis in Rᵖ, then the members of S span Rᵖ and hence form an orthogonal set.