Number of subspaces of a vector space over a finite field

[winter85]

Homework Statement

Prove: If V is an n-dimensional vector space of a finite field, and if 0 <= m <= n, then the number of m-dimensional subspaces of V is the same as the number of (n-m)-dimensional subspaces.

The Attempt at a Solution

Well here's a sketch of my argument. Let U be an m-dimensional subspace of V, then the annihiator of U, U^0 is a (n-m)-dimensional subspace of V*, the dual space of V. Let W be the subspace of V whose dual space is U^0. I plan to show that W is in 1-to-1 correspondence with U, so there is an injection between the set of m-dimensional subspaces of V and the set of (n-m) dimensional subspaces. Since the situation is symmetric, it follows those sets have a bijection and therefore the same cardinality.

Now before I work out the details, I want to ask about one thing, this argument nowhere uses the fact that V is a vector space over a finite field (except perhaps at the very last, to substitute "cardinality" by "number of elements"). So is there something wrong with it? why is the problem specifically about vector spaces over finite fields if it works in the general case?

Thanks.

 

[mighty2000]

Thank you for your post. Your approach to proving this statement seems sound. However, you are correct in noting that the fact that V is a vector space over a finite field is not explicitly used in your argument. This is because the statement holds true for any vector space, regardless of the underlying field. However, the reason the problem specifies finite fields is because the concept of dimensionality and subspaces is slightly different in finite fields compared to infinite fields.

In finite fields, the concept of dimensionality is slightly different because the field itself has a finite number of elements. This means that in a finite-dimensional vector space, the dimension is also finite. This is in contrast to infinite fields, where the dimension can be infinite.

Furthermore, in finite fields, the number of elements in a subspace is also finite. This means that the number of m-dimensional subspaces and (n-m)-dimensional subspaces in a finite-dimensional vector space is also finite. This is not necessarily the case in infinite fields.

Therefore, while your approach to proving the statement is valid in both finite and infinite fields, the specific mention of finite fields in the problem is to highlight the slightly different concept of dimensionality and subspaces in these fields.

I hope this helps clarify why the problem specifies finite fields. Keep up the good work in your research!