Is the Union of W and Its Orthogonal Complement Equal to V?

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Discussion Overview

The discussion revolves around the relationship between a vector space V, a subspace W, and the orthogonal complement of W. Participants explore whether the union of W and its orthogonal complement equals V, examining implications for vector space properties and subspace definitions.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that the union of W and its orthogonal complement does not equal V, citing that the union of two vector subspaces is not generally a subspace.
  • Another participant agrees with this view, stating that the union would not be a subspace of V unless W is either the zero vector or the entire space V.
  • It is noted that the direct sum of W and its orthogonal complement equals V, contrasting with the union's properties.
  • A participant references an external page for hints regarding the proof of the union's relationship to V, though the content of that page is not discussed.

Areas of Agreement / Disagreement

Participants generally disagree on the claim that the union of W and its orthogonal complement equals V, with multiple viewpoints presented regarding the properties of unions and direct sums in vector spaces.

Contextual Notes

The discussion does not resolve the mathematical steps or definitions involved in the claims about unions and direct sums, leaving some assumptions and implications unaddressed.

xfunctionx
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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-
 
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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.
 
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.
 

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