Do Orthogonal Subsets Spanning the Same Subspace Prove Equality?

[NATURE.M]

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 there's a better method, I'd like to know.
Thanks!

 

[Office_Shredder]

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.

 

[NATURE.M]

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]

OK then yeah you are doing more work than required. If they're orthogonal sets you know their spans have dimension equal to the number of elements. As soon as you have one span is contained in the other you are done, and you don't need to check the other direction.

 

[NATURE.M]

okay that makes sense. thanks!