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

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In summary, the conversation discusses the properties of orthogonal complements in vector spaces. It is proven that the orthogonal complement of a subspace is also a subspace, and the intersection of a subspace and its orthogonal complement is zero. However, it is noted that the union of a subspace and its orthogonal complement is not equal to the original vector space, unless the subspace is either the zero vector or the entire vector space. Instead, the direct sum of a subspace and its orthogonal complement is equal to the original vector space.
  • #1
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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  • #2
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
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
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.
 

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

What is an orthogonal complement?

An orthogonal complement is a mathematical concept that refers to the set of all vectors that are perpendicular, or orthogonal, to a given vector or set of vectors.

How is the orthogonal complement denoted?

The orthogonal complement of a vector or set of vectors is denoted as a superscript with a perpendicular symbol, such as U⊥ for the orthogonal complement of vector U.

What is the relationship between a vector and its orthogonal complement?

A vector and its orthogonal complement are always perpendicular to each other, meaning that their dot product is equal to 0.

How is the orthogonal complement useful in linear algebra?

The orthogonal complement is useful in linear algebra for several reasons, including finding the basis of a subspace, solving systems of linear equations, and understanding the geometry of vector spaces.

Can the orthogonal complement be applied to any vector space?

Yes, the concept of orthogonal complement can be applied to any vector space, as long as the space has a defined inner product or dot product operation.

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