linear algebra / higher dimensional geometry

[thepillow]

Hi,

I'm trying to get a better feel for how different k-flats can intersect in higher dimensions. It's clearly impossible to visualize this once we get past the 3-dimensional physical space that we can see, but how would I go about answering a question like this:

Let's say we have a 3-flat in 7 dimensions, i.e. we're in R⁷. What's the largest possible dimension that another flat can have if it is to intersect our 3-flat in exactly one point? What if it is to intersect our 3-flat in a 2-flat?

If I'm given an equation for a flat and told that it intersects a k-flat in exactly a point (or a line, etc) I know how to find equations to describe this. I'm just having trouble with the more abstract questions of what types of intersections are possible.

Thanks in advance for the help!

 

[daon2]

Consider that the intersection of two subspaces U,V of R^n has possible dimensions: 0,1,...,min{dimU,dimV}. Their intersection is (dimension) 0 if they only intersect at the origin. If dimU=3 as in this case, take a basis b1,..,b3 of U and extend it to a basis b1,..,b3,a1,a2,a3,a4 of R^7. Then V=span{a1,a2,a3,a4} intersects U trivially. What if dimV > 4?

 

[thepillow]

Consider that the intersection of two subspaces U,V of R^n has possible dimensions: 0,1,...,min{dimU,dimV}. Their intersection is (dimension) 0 if they only intersect at the origin. If dimU=3 as in this case, take a basis b1,..,b3 of U and extend it to a basis b1,..,b3,a1,a2,a3,a4 of R^7. Then V=span{a1,a2,a3,a4} intersects U trivially. What if dimV > 4?

Thanks daon2!

So, in other words, regardless of the value of n, a 3-flat and a 2-flat can intersect in a flat of dimension no greater than min{2,3}= 2, right? That makes sense that we're limited to the dimension of our lower-dimensional flat.

I'm still a bit stuck though in trying to get the general idea about how the particular dimension n of the ambient space affects the possible intersections we can have within it between flats of different dimensions. Let me take my hypothetical example and make it concrete. Let's say we're in R⁷ and we have a 3-flat given by this vector-parametric representation (I just picked some really simple numbers to work with):



Now let's say that we want to find a flat of the largest possible dimension that intersects our 3-flat in a 2-flat. How can I think about this question (somewhat) intuitively?

There obviously some things we can rule out without much thought like, for example, an 11-flat clearly doesn't work since we're only in R⁷, and neither does a 7-flat, since that's the whole space, etc... But these are the trivial examples.

I'm having trouble because my linear algebra professor really wants us to have an intuitive grasp for these things, but unfortunately it's very hard for him to explain them. I've been looking online to try to find some stuff to read on this topic, but I can't really find anything that talks about this particular concept (flats intersecting in various dimensions) in a way that's understandable to someone in their first linear algebra course.

thanks a lot!

 

[daon2]

Ah, I assumed a "k-flat" was a k-dimensional subspace of R^n. But it looks like I was wrong if your example above is one. However, i believe the idea is similar:

In this case, your 3-flat, say T, is generated by p+t1*e5+t2*e6+t3*e7. Note that the vectors p+e1,...,p+e4 do not belong to your 3-flat. So any flat intersecting T in a 2-flat can contain these, as well as any two linearly independent vectors from T. So: 4+2=6.

So, if I interpreted the question correctly this time, an example of a largest-dimension 6-flat intersecting T in a 2-flat is p+a1e1+a2e2+a3e3+a4e4+t1e5+t2e6, where ai, ti are parameters. Their intersection is, of course, {v in T | t3=0}

 

[thepillow]

Ah, I assumed a "k-flat" was a k-dimensional subspace of R^n. But it looks like I was wrong if your example above is one. However, i believe the idea is similar:

In this case, your 3-flat, say T, is generated by p+t1*e5+t2*e6+t3*e7. Note that the vectors p+e1,...,p+e4 do not belong to your 3-flat. So any flat intersecting T in a 2-flat can contain these, as well as any two linearly independent vectors from T. So: 4+2=6.

So, if I interpreted the question correctly this time, an example of a largest-dimension 6-flat intersecting T in a 2-flat is p+a1e1+a2e2+a3e3+a4e4+t1e5+t2e6, where ai, ti are parameters. Their intersection is, of course, {v in T | t3=0}

Hey daon2,

I think you definitely interpreted the question correctly. Thanks a lot for the help.