Linear Algebra: Shilov. Hyperplanes

1. Oct 28, 2008

mrb

NOTE: I may have figured this out; read the end of the post....

From Shilov's Linear Algebra, Chapter 2, pg.57, #13.

1. The problem statement, all variables and given/known data

The hyperplanes H_1 and H_2 have dimensions p and q respectively. What is the smallest dimension which the hyperplane H_3 must have in order to be sure to contain both H_1 and H_2?

2. Relevant equations

I'm unsure if Shilov's hyperplane is a standard concept, so I'll describe it here. It is basically a coset, regarding the vector space as a group under addition:

Let L be a subspace of a linear space K, and let x in K be a fixed vector. Let H be the set of all vectors of the form x + y for all y in L. Then H is a hyperplane. Its dimension is the dimension of L.

3. The attempt at a solution

Shilov's answer is p + q + 1, if this does not exceed the dimension of the space. But how can this be right? We can regard any subspace as a hyperplane, because we can use 0 as the value x from the definition.

So in R^4 let H_1 and H_2 both be the subspace with basis (1, 0, 0, 0). Then they are both of dimension 1, so p+q+1 = 3, and the subspace with bases (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1) doesn't contain either of them but is of dimension 3!

WELL, after typing that out I just realized what Shilov may have been getting at. I took him to mean: what is the minimum dimension some arbitrary H_3 must be that we know it contains both? But perhaps he meant: What is the minimum dimension such that some H_3 of that dimension contains both?

I still think my interpretation of the question is the most natural and I'm not completely certain the other is what he meant. What do others think?

Thanks.

Last edited: Oct 29, 2008
2. Oct 28, 2008

Dick

I think your realization at the end is getting at what Shilov must have meant. If H1 has dimension p and H2 has dimension q then they are contained in an H3 of dimension AT MOST p+q+1. Is that what you are saying? And, yes, I don't think his or her use of the term 'hyperplane' is very standard. I've only used hyperplane used to describe a linear set of dimension n-1 in a space of dimension n. I would call it those 'affine subspaces'.