# Subspaces and perpendiculuar subspaces

## Homework Statement

How do you show that M double perp is a subset of M?

## The Attempt at a Solution

My prof told me to try proving that M is a subset of M perp perp, then to use the facts that if M is a subspace of Rn then T(X) = projU(X) for all X in Rn.

I'm not sure how to go about that. I know logically that it's a subset, but I don't know how to prove it.

I'm thinking that once I prove it, maybe I can show that the dimension of U and U perp perp are equal, so the spaces are equal too?

Can anyone help get me started? Thanks :)

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Office_Shredder
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It follows pretty much from the definition of perpendicularity. Try it out and see what happens

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Okay well I tried this:
Say X is a vector in M.

Then Y is a vector in M perp if Y ● X = 0.

Z is a vector in M perp perp if Z ● Y = 0.

But I can show that every vector X in M is also in M perp perp:
X ● Y = 0 because Y ● X = 0.

Therefore M is a subset of M perp perp. Is that part right?

Office_Shredder
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Yeah, that's all there is to it for that part

Okay now for the next part.
T(X) = projM(X) = Y
dim(M) + dim (M perp) = n

S(Y) = projM perp(Y) = Z
dim(M perp) + dim(M perp perp) = n

Then n - dim(M) = n - dim(M perp perp)
dim (M) = dim(M perp perp)

Since M is a subset of M perp perp then they are equal?

Office_Shredder
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I'm not sure precisely what your projection notation is supposed to say but the argument basically goes like how you posted:

For all subspaces U, dim(U)+dim(U perp)=n

So dim(M)+dim(M perp)=n

M perp is a subspace also, so dim(M perp)+dim(M perp perp)= n

And then subtract like you did to finish it off

Hmm okay, so the projection thing isn't actually necessary?

Thanks for your help, by the way!