# A LinAlg Proof Involving Orthogonal Complement

1. May 6, 2012

### jpcjr

1. The problem statement, all variables and given/known data

Here is the problem and my complete answer.

Am I OK?

Thanks!

http://www.d-series.org/forums/members/52170-albums1546-picture8143.jpg [Broken]

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: May 6, 2017
2. May 7, 2012

### 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) $S\subseteq S^{\bot \bot}$.

3. May 8, 2012

### 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