Can it happen that a non-trivial subspace of a vector space V (i.e., a subspace different from 0 and V) has a unique complement?
My answer to this would be no in general, but I do not know if there exists some exotic subspace for which this holds.
Here's my attempt at some sort of a proof:
Let K and H be subspaces of V.
Let be a basis of K.
Let be a basis of H.
K and H are complementary subspaces if and only if the complete set,
is a basis of V.
Say now that I have K and want to find a complementary subspace. Since every set of linearly independent vectors may be used as a basis, H is not unique.
I can't understand how you may think that you've proved anything...What you now must prove is that no matter what H is (non-trivial, of course), complementing its basis to a basis of the whole space cannot be made in one unique way, and this is true ALWAYS...and here you may want to distinguish different fields: finite and infinite ones.
Any suggestions are greatly appreciated.