Othogonal complement of a span

[jOc3]

Homework Statement

Show that <S>^{\bot}=S^{\bot}

Homework Equations

The Attempt at a Solution

I manage to show S^{\bot}\subseteq<S>^{\bot}.
What about the other way round? Any way of proving without using the concept of basis?

 

[siddharth]

You need to prove that span(S)^{\bot} is a subset of S^{\bot}. However, you know that S \subseteq span(S). So, given the latter, try to prove the former.

 

[jOc3]

This is what I come out with:

Let y\in<S>^{\bot}, then <y,w>=0, for all w\in<S>.
But S\subseteq<S>.
Hence for y\in<S>^{\bot}, then <y,w>=0, for all w\inS.
Hence y\inS^{\bot}.
Hence <S>^{\bot}\subseteqS^{\bot}.

Is it correct? Or should I say "for some w"?