Talk:Frame of a vector space

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From the section Relation to bases:

Consequently, a frame is a set of vectors which [...] cannot be any arbitrary set of vectors which spans V.

That sentence doesn't make any sense to me at all. Read on its own, it may be construed to mean that no set of vectors which spans V is a frame, which of course is wrong. (Only sets which span V can possibly be frames.) I know it wasn't meant to be read that way, but the wording seems very ambiguous and unclear to me. Does someone feel up to improving that part? Eriatarka 10:57, 28 April 2007 (UTC)

The sentence is meant to point out the fact that you cannot take an arbitrary set of vectors which span a vector space and call them a frame of that vector space, ie, a frame is not just a simple generalization of a basis where the linearly independence has been dropped. The 'frame condition' is more strict than just dropping linearly independence. The set of vectors presented before that sentence is an example of a set which span a vector space but does not satisfy the 'frame condition' and therefore is not a frame. I don't understand in what way the sentence is ambiguous, can you please give an example? --KYN 07:49, 29 April 2007 (UTC)

I read it as ambiguous in the sense that I mentioned above:

>>Read on its own, it may be construed to mean that no set of vectors which spans V is a frame, which of course is wrong.<<

I would simply formulate the statement as "Not every set of vectors which spans V is a frame of V." or "Spanning the space V is not a sufficient condition for being a frame." or maybe "There are sets of vectors spanning V which are not frames of V." These all seem much clearer to me. Eriatarka 17:53, 29 April 2007 (UTC)

The suggested formulations are all fine to me --KYN 19:32, 29 April 2007 (UTC)

[edit] The word "frame"?

A three-dimensional coordinate system. A frame is an ordered set of basis vectors, here labeled x, y, and z. The vectors can be thought of as "framing" the vector space.

I like having a sense of why mathematical terms are called what they are. I'm thinking of adding this picture and caption regarding the intuition of the word "frame".

Is that accurate? Could it be phrased better. Comments? —Ben FrantzDale 01:36, 3 May 2007 (UTC)

The term frame is used in mathematics in at least two different ways. One is the meaning related to a coordinate frame or (more commonly) a coordinate system, which is what I believe that your are thinking about here. Another is the meaning described in this article which can be described as a specific way of generalizing a basis (spans a space and is linearly independent) to something which is spans the space but is not required to be linearly independent. Instead it must satisfy the frame condition described in the article. These two concepts are quite different and should not be confused. The picture you have posted here appears to be related to the first meaning of frame but not (in a way that I understand) related to the second meaning. Personally, I try to avoid using frame in the first sense since there already is a well-established term coordinate system which can be used instead and which (I believe) is more common. --KYN 10:14, 3 May 2007 (UTC)

The term frame used in the sense described in the article way, as far as I understand it, coined by Duffin & Schaeffer in their article listed in the references. Is see now that the Frame article has changed its description of this article. I will need to change it back again. --KYN 10:14, 3 May 2007 (UTC)

OK, let me try to get an intuitive understanding. Would it be accurate to say it's called a "frame" because it is a thing on which a vector space is built (akin to the frame of a building)?

Sorry, no idea why they called this particular construction a frame. The similarity with a frame of a building sounds plausible but I have also noticed that there are other meanings of that word which may fit; frame as in boundary of something.

If the definition is basically just "it's like a basis with the possible addition of 'extra' vectors that are linear combinations of the basis vectors", then would it be reasonable to say that in the 2-D image I posted above, treating it as a 2-D image, the vectors x, y, and z form a frame for R²?

Yes, this is correct.

A non-example given is

{ (1,0) , (0,1), (0,1) , (0,1) ... }

but that's not a set. that's a multiset. I think a set that has similar properties would be

{ (1,0), (0,1), (0,1.5), (0,1.75), ... }. Am I right that that set is not a frame?

The multiset idea is, I believe, a matter of taste. As far as I understand it, you need a set of vector (elements of the vector space) which are enumerable and if some of them are identical or not is not a big deal. But you are right that this last set is not a frames since the upper frame bound B = infinity.

Would the set

{ (1,0), (0,1), (0,1/2), (0,1/4), (0,1/8),... }

be a frame for R²? Those vectors do span the space but aren't linearly independent. However, they do seem to satisfy the frame condition.

Yes, this is an example of a set of vectors which constitute a frame. But is doesn't have to be infinite. These two sets of vectors in R² are also frames

{ (1,0), (0,1), (0,1/2) }

{ (1,0), (0,1)}

Thanks. —Ben FrantzDale 12:07, 3 May 2007 (UTC)

I believe that the essential idea behind this particular generalization of bases to sets which are not linearly independent is that you want to be able to construct a dual frame, i.e., a dual set of vectors such that if you have the scalar products of a vector v relative to the frame vectors, you can reconstruct v by linear combinations of the dual frame and the scalar products. This is in analogue to the relation between a basis and its dual (or reciprocal) basis. What Duffin & Schaeffer did was to prove that the frame condition is a sufficient (I don't know about necessary) condition for a dual frame to exist. --KYN 22:00, 3 May 2007 (UTC)

As I understand any finite set which include basis is a frame. Does that mean that only infinite frames are really interesting ? Or for finite frames object of interest is exact bounds A and B ? Serg3d2 06:03, 8 July 2007 (UTC)

The practical use of frames is not limited to the infinite dimensional cases. In a sense, frames introduce redundancy, regardless of whether the dimensionality is finite or infinite, which can be used, e.g., for noise reduction in signal processing. Oversampling is a good example of how frames can be used precisely in this way and it works also for a finite set of samples. --KYN 10:25, 8 July 2007 (UTC)