# Orthogonal Complements.

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

CompuChip
Science Advisor
Homework Helper
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.

matt grime
Science Advisor
Homework Helper
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.

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.