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!

 

[Hurkyl]

Please don't double post

 

[M98Ranger]

Hi there,

The intersection of two subspaces in an N-dimensional space can be visualized as the common points or elements that both subspaces share. In general, the dimension of the intersection of two subspaces will depend on the dimensions of the subspaces themselves and the dimension of the overall space.

Let's start by considering the case where N = 3. In this case, both subspaces U and W have a dimension of 2. This means that they are both planes in a 3-dimensional space. Now, imagine two planes intersecting in 3-dimensional space. The intersection of these planes will form a curve, as you mentioned. This is because the intersection of two planes in 3-dimensional space is a line, and a line can be represented as a curve in 3-dimensional space.

Moving on to the case where N = 4, we have the same subspaces U and W with a dimension of 2. However, now they are both planes in a 4-dimensional space. In this case, the intersection of the two planes will form a finite number of points. This is because in 4-dimensional space, the intersection of two planes is a line, and a line can be represented as a finite number of points.

Now, for the case where N > 4, the subspaces U and W with a dimension of 2 are still planes, but they are now in a space with more than 4 dimensions. In this case, the intersection of the two planes will not necessarily intersect at all. This is because in higher dimensions, there is more space for the planes to be positioned in a way that they do not intersect.

To prove this mathematically, we can use the concept of linear independence. A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the others. In other words, no vector in the set can be expressed as a linear combination of the other vectors in the set.

Now, if we consider the two subspaces U and W, each with a dimension of 2, we can represent them as two sets of linearly independent vectors. In order for the subspaces to intersect, there must exist a vector that can be written as a linear combination of the vectors in both sets. However, in higher dimensions (N > 4), there is more space for the vectors to be positioned in a way that they are linearly