Basis and Subspace Help: Exploring the Relationship between Vectors e and d

[Sue_2010]

Suppose I have 3 vectors e1, e2, e3 that spans the subspace E, another 3 vectors d1, d2, d3 that spans the subspace D. If I also know that e1’d1 = 0, e2’d2 = 0, e3’d3 = 0, are there any conclusions I can make in terms of E and D? like row(E) = null(D)?

 

[rochfor1]

what is row(E) and null(D)...E and D are subspaces, not matrices. If you mean E=[e1 e2 e3] and D=[d1 d2 d3], you can't really make any judgements, to conclude that row(E)=null(D) you'd have to know that ei'dj=0 for all i,j=1,2,3.

 

[rochfor1]

Actually, I was wrong. Even then all you would get is that $$D \subset \operatorname{null} E^\top$$ and similarly $$E \subset \operatorname{null} D^T$$. The questions of whether or not we may even have row(E)=null(D) depends on the ambient space, because in general the dimensions of these two subspaces need not agree even if E and D are both 3-dimensional. Apologies for the mix up.

 

[Sue_2010]

Thank you for your reply!

You are right, E and D are matrices. Let me state my question more clearly.

I want to find a projection that maximally preserves the Euclidean distance between two points (vectors in general). Consider a 2-D example as show in the figure. The direction that maximally preserves the distance between c1 and c2 will be (c2-c1), or I can say it's the row space for matrix E = [(c2-c1)].
http://www.personal.psu.edu/sxy162/temp/1.jpg

If c2 and c1 have the same norm, then (c2+ c1) is perpendicular to (c2 - c1), or (c2-c1)*(c2+c1)' = 0. Therefore, instead of projecting to row of matrix E = [(c2-c1)], I can project to the null space of matrix D = [(c2 + c1)], right? It only works if all the vectors are normalized.

Now my question is about more general problem. Say
E = [
(c1,1 - c1,2)
(c2,1 - c2, 2)
...
(cn,1 - cn, 2)
]

and

D =
[
(c1,1 + c1,2)
(c2,1 + c2, 2)
...
(cn,1 + cn, 2)
]

Now, matrices E and D each has n rows, instead of 1 row in the first example.

The question is, can I still say that, projecting to null(D) is equivalent to projecting to row(E)?

 

[Sue_2010]

Hi rochfor1,

Now I understand that it's not right to compare null(D) and row(E), since they are of different dimensions.

Can we compare the projections to null(D) and row(E) in terms of their effects to the separability of the pairs of vectors, (c11 , c12) ... (cn1 , cn2) I mentioned in the above thread?

 

[Sue_2010]

And how do you get D in null(E') and E in null(D') ?

 

[Sue_2010]

Hello everyone,

I really want to get some feedback. Thank you!