Orthogonal Complements.

  • Thread starter xfunctionx
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  • #1

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Hi, I was just reading about Orthogonal complements.

I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V.

I then proved that the intersection of W and its orthogonal complement equals 0.

However, I am wondering if the union of W and its orthogonal complement equals V?

Can anyone please answer that, and if so, can you give a proof?

Thanks.

-xfunctionx-
 

Answers and Replies

  • #2
CompuChip
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It is true, says see this page. The links on the page will give you some hints as to in which direction the proof should be found.
 
  • #3
matt grime
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The union is not V: the union of two vector subspaces is not in general a subspace: just remember that R^2 is not the union of two lines.

V is the vector space sum of W and its complement.
 
  • #4
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As Matt Grime said, the union is not V. The union would not even be a subspace of V, unless W = {0} or W = V. However, the direct sum of W and its orthogonal complement is equal to V.
 

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