Intersecting subspaces in N dimensions

[pat_connell]

Hey guys,

I have a little problem here:

given two subspaces U and W both of dimension two of an N dimensional space show in general that if N = 3 the intersection of U and W forms a curve; if N = 4 a finite number of points; and N > 4 they do not in general intersect at all.

I can kind of visualize the answer for the first two cases based on analagous cases lines viewed in R2 and R3, but I am not really satisfied with the answers.

I would like a really rigorous maths proof.

please help!

 

[matt grime]

Are yuo talking vector (sub)spaces here or something else?

I'll assume vector space since you're talking about dimension.

The interesection of two vector subspaces is a vector subspace of each. So the interesection is going to be either 0,1, or 2 dimensional irrespective of the ambient space, and thus the alleged curve is in fact a straight line, the "finite number of points" is in fact a single point

Now, all you need to do is formally tell us what you mean by space subspace and generally, ie do you mean picked at random where random has sort of greater meaning? Then we might be able to give a better answer.

 

[pat_connell]

Yes I am talkig about vector subspaces and vector spaces and possibly even topological spaces.

But its alright i think I've figured out the answer