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Subspaces.
Hey again!
Sorry, I don't mean to fill up the entire forum, but as I said in my other post, I have an exam and I need all the help I can get. I figured if anyone can answer these for me while I study, bonus! Here's a few questions I could use the answers for.
Attachment 21397
I put them as pictures because they're much more clear that way than if I tried to type them.
Again, thank you to whoever is willing to help! :)

For 1:
$\displaystyle x \in U \Rightarrow <x, U^\perp> = 0 \Rightarrow x \in \left( U^\perp\right)^\perp $.
So we have $\displaystyle U\subseteq \left( U^\perp\right)^\perp^{(1)}$.
$\displaystyle dim_\mathbb R U = dim_\mathbb R U^\perp, dim_\mathbb R U^\perp= dim_\mathbb R \left( U^\perp\right)^\perp \Rightarrow dim_ \mathbb R U= dim_\mathbb R \left( U^\perp\right)^\perp ^{(2)}$.
$\displaystyle \overset{(1),(2)}{\Longrightarrow} U = \left( U {^\perp}\right)^{\perp}$.

For 2 :
(i)
$\displaystyle x\in (U+V)^{\perp}\Rightarrow \ldots$
Next step?
(ii)
$\displaystyle x\in U^{\perp}\cap V^{\perp}\Rightarrow \ldots$
Next step?

two more hints:
if <x,u> = 0, and <x,v> = 0 then certainly <x,u+v> = 0.
now can we say that if u is in U, and v is in V, that u,v are in U+V?