Sunday, August 27, 2006
a problem
Heres an interesting linear algebra problem: Let F be a field and let V be a finite dimensional vector space over F. If α1 , ... , αm are finitely many vectors in V, each different from zero vector, prove that there is a linear functional f on V such that
f(αi) ≠ 0 , i = 1 , .., m
My attempts at proving this lead to this problem: Can a union of finitely many sub spaces of dimension N-1 (hyperplanes) cover a vector space of dimension N.? Which seems an interesting problem in itself.
f(αi) ≠ 0 , i = 1 , .., m
My attempts at proving this lead to this problem: Can a union of finitely many sub spaces of dimension N-1 (hyperplanes) cover a vector space of dimension N.? Which seems an interesting problem in itself.
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You would have figured out this by now, kyon ki Hahn ne Banache theorem, kiya tera problem kam!
Or are you second time puzzled?
Or are you second time puzzled?
# posted by 
: 1:42 AM
: 1:42 AM 
Now the result appears to be a consequence of Baires theorem which can be seen as follows:: The union of subspaces of, formed by the annihilators of each of non-zero vectors in the fdvs X , being proper subspaces, cannot equal the whole space of functionals on X, since each subspace is nowhere dense and closed. So there must exist some functional which does not annihilate each of these non-zero vectors. Back then I had to take a more elaborate route of proving that the union is proper by actually trying to construct one a functional outside of the union.. I think a similar problem exists somewhere in Linear algebra problem book by Halmos... It would be interesting to see a solution using Hahn banach....
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