Common complementary vector subspaces

[Mathmos6]

Homework Statement

Show that any two subspaces of the same dimension in a finite-dimensional vector space have a common complementary subspace. [You may wish to consider first the case where the subspaces have dimension 1 less than the space.]

The Attempt at a Solution

I've managed to sort out the case where the subspaces have dimension 1 less than the space I believe, using the first part of the question: "let U be a subspace of F^n. Show that there is a subset I of {1,2,...,n} for which the subspace W = span({e_i : i ∈ I}) is a complementary subspace to U in F^n." However, i tried using induction on the general case for k=dim(W) (so we've sorted the k=1 case if my proof is correct) and things got very messy and long. Could someone help me out with a more concise/neat solution which doesn't use too much complicated machinery?

Thanks a lot :)

 

[mighty2000]

Hello! I believe I have a simpler solution for this problem.

First, let us consider the case where the subspaces have dimension 1 less than the space. Let U and V be two subspaces of F^n with dim(U)=dim(V)=n-1. Since dim(U)=n-1, U cannot contain the entire F^n, so there exists a vector u in F^n that is not in U. Similarly, there exists a vector v in F^n that is not in V.

Now, consider the subspace W=span({u,v}). Since u and v are not in U and V respectively, it follows that W is not contained in U or V. This means that W is a complementary subspace to both U and V in F^n.

Now, let us consider the general case where U and V have the same dimension k. We can use induction on k to prove that there exists a complementary subspace W to both U and V in F^n.

Base case: k=1
This is the case we have already proved above.

Inductive step: Assume the statement is true for subspaces of dimension k-1. Now, let U and V be two subspaces of F^n with dim(U)=dim(V)=k. Consider the subspaces U' and V' where U'=U∩W and V'=V∩W, where W is the complementary subspace we constructed in the base case.

Since dim(U)=dim(V)=k, it follows that dim(U')=dim(V')=k-1. By our assumption, there exists a complementary subspace W' to both U' and V' in W.

Now, consider the subspace W''=W+W'. Since W' is a complementary subspace to U' and V' in W, it follows that W'' is a complementary subspace to U and V in F^n.

Therefore, by induction, we have proved that any two subspaces of the same dimension in a finite-dimensional vector space have a common complementary subspace.

I hope this helps! Let me know if you have any questions or if you would like me to clarify anything. Good luck with your studies!