Orthogonal Complements.

[xfunctionx]

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-

 

[CompuChip]

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]

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.

 

[Werg22]

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.