Finding the orthogonal projection of a vector without an orthogonal basis

In summary, the conversation discusses a result in euclidean spaces where a subspace of finite dimension can be represented as a span of vectors. The result states that for any element x in the space, its orthogonal projection onto the subspace is equivalent to the inner product of x and the vectors in the subspace being equal to 0. The post also mentions that this result is more general than just for orthogonal families of vectors.
  • #1
AimaneSN
5
1
Hi there,

I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove :

Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E##

Then we have:

##\forall y \in F## : ##y=p_F(x) \Leftrightarrow \forall i= 1,...,p : (x-y,e_i) = 0##

where ##(.,.)## denotes an inner product
and the linear map ##p_F## is the orthogonal projection onto ##F##.

I managed to prove the equivalence only when the family of the vectors ##(e_i)## is orthogonal but the result is more general.

Thank you and I would appreciate any hint or help.
 
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  • #2
The inner product ##(x-y,e_i)## is zero for all ##i## if and only if ##x-y\in F^\perp## if and only if ##x## is of the form ##x=y+z## where ##z\in F^{\perp}## if and only if ##y## is the orthogonal projection of ##x## onto ##F##.

Your post was a bit of effort to read- it would be better to use latex.
 
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Likes AimaneSN
  • #3
Thank you for your reply, it's much clearer now. I just modified my post using Latex code.
 
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Likes WWGD
  • #4
OP: Notice you can always use Gram-Schmidt to orthogonalize your basis vectors, and this won't affect the span .
 
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Related to Finding the orthogonal projection of a vector without an orthogonal basis

What is an orthogonal projection of a vector?

An orthogonal projection of a vector is a process of finding a vector that is perpendicular to a given vector and lies on a given subspace.

Why is an orthogonal projection important in scientific research?

An orthogonal projection is important in scientific research because it allows us to break down a complex vector into simpler components, making it easier to analyze and understand.

What is an orthogonal basis?

An orthogonal basis is a set of vectors that are mutually perpendicular and span a given subspace. In other words, each vector in the basis is orthogonal to all other vectors in the basis.

Can a vector have an orthogonal projection without an orthogonal basis?

Yes, a vector can have an orthogonal projection without an orthogonal basis. This can be achieved by using the Gram-Schmidt process to find an orthonormal basis for the subspace that the vector is being projected onto.

How do you find the orthogonal projection of a vector without an orthogonal basis?

To find the orthogonal projection of a vector without an orthogonal basis, you can use the formula:
projuv = (v · u) / (u · u) * u
where u is a unit vector in the direction of the subspace and v is the vector being projected.

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