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$$\in$$S.
Hence y$$\in$$S$$^{\bot}$$.
Hence <S>$$^{\bot}$$$$\subseteq$$S$$^{\bot}$$.

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