A LinAlg Proof Involving Orthogonal Complement

[jpcjr]

Homework Statement

Here is the problem and my complete answer.

Am I OK?

Thanks!

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Homework Equations

The Attempt at a Solution

 

[micromass]

No, it is not ok. You seem to prove that if $u,v\in S^\bot$, that cu+v is an element of S. But this is simply not true at all.
Likewise, you take $a,b\in S^{\bot \bot}$ and you conclude that these are in $S^\bot$. But this is also not true.

In short, it is NOT true that

$$S^{\bot \bot}\subseteq S^\bot \subseteq S$$

How do you prove the theorem, well you need to prove two things:

1) $S^{\bot \bot}$ is a subspace.
2) $S\subseteq S^{\bot \bot}$.

 

[jpcjr]

Thank you!

By the skin of my teeth, some help from you, and the grace of God, I received the best grade I could have expected in Linear Algebra.

Thanks, again!

Joe