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Topic: Generalizing Orthogonal Projection
Replies: 7   Last Post: Jan 24, 2013 5:17 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Generalizing Orthogonal Projection
Posted: Jan 22, 2013 10:33 AM
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On Mon, 21 Jan 2013 21:47:11 -0800 (PST), wrote:

>Hi, All:
>I'm just curious about wether orthogonalprojection generalizes to cases such as
>Say we have a 1-D subspace L (i.e., a line thru the origin) in R^3 , and
>let q=(x,y,z) be a point in R^3 which is not on the line. Then I don't see
>how to project q orthogonally onto L; I can see how to project q into a
>2-D subspace P : the projection of q into P is the ortho complement, and
>every vector in P is in the orthogonal complement of the ortho projected
>line. But, the same is not the case with q and L. Sorry for the rambling;
>my question is then actually:
> If L is a 1-D subspace of R^3, and q=(x,y,z) is a point not on L. Can we
>define the orthogonal projection of q into L, or do we need to have a plane
>P (as subspace) , to define an ortho projection of q?

Yes. The projection is p, defined by two conditions:

(i) p lies on L

(ii) q - p is orthhogonal to L.

Or: If V is any subspace of R^n, let W be the orthogonal
complement: W is the space of all w such that

v.w = 0 for all v in V

(or in another standard notation, <v, w> = 0).

Then every x in R^n has a uniquue decomposition

x = v + w

where v is in V and w is in W; now v is the orthogonal
projection of x onto V.

Yes, you could also define this in terms of certain planes,
but that's sort of missing the point.


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