Let B be a skew symmetric bilinear form on an n-dimensional vector space V over

Question

Let B be a skew symmetric bilinear form on an n-dimensional vector space V over R.
Together, we will show that there is a basis u1,..., uk, e1,...,em,f1,...,fm of V such that
1. B(ul,v)=0 for each l=1,...,k and all v?V
2. B(ei,ej) = 0 = B(fi,fj) for any i, j = 1,...,m

3. B(ei,fj) =1 if i=j and B(ei,fj)=0 if i =j.

Problem 5.2. Show that, based on the definition of U and the fact that W is a comple-
mentary space to U, there must be another vector in W, call it F1 such that B(e1,F1) =
0. (Hint: Assume otherwise so that B(e1,w) = 0 for all w ? W. Show why this leads
to a contradiction and so to the result.)
Thus, B(e1, F1) = c for some nonzero scalar c ? R. If c = 1, then simply define f1 to be F1;
otherwise, if c = 1, define f1 = 1F1.

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