# Proving the subspaces are equal

If I want to show two orthogonal subsets S$_{1}$ and S$_{2}$ of ℝ$^{n}$ both span the same subspace W of ℝ$^{n}$ does it suffice to show that
S$_{1}$$\subset$S$_{2}$ and that S$_{2}$$\subset$S$_{1}$, thus showing S$_{1}$ = S$_{2}$
$\Rightarrow$ they span the same space.

If theres a better method, I'd like to know.
Thanks!

Office_Shredder
Staff Emeritus
Gold Member
Yes, that method would work if the two sets are equal but that will almost never be the case. Typically you would want to show that
$$S_1 \subset span(S_2)$$
which immediately implies
$$span(S_1) \subset span(S_2)$$
at which point since they both have the same size (if they don't then you didn't need to do any work) the two spans must be equal.

Yes, that method would work if the two sets are equal but that will almost never be the case. Typically you would want to show that
$$S_1 \subset span(S_2)$$
which immediately implies
$$span(S_1) \subset span(S_2)$$
at which point since they both have the same size (if they don't then you didn't need to do any work) the two spans must be equal.
After looking back at my post, I realize I should of wrote span(S$_{1}$) $\subset$ span(S$_{2}$) and vice versa. But anyways thanks.

Office_Shredder
Staff Emeritus