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Vector Spaces, Basis and Quotient Spaces

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See the attached file.
1. Let and be vector spaces over and let be a subspace of

Show that for all is a subspace of
and this subspace is isomorphic to .

Deduce that if and are finite dimensional, then
dim = (dim - dim )dim

2. Let be a linear operator on a finite dimensional vector space
Prove that there is a basis of such that is a
basis of im and is a basis of ker

3. Let and be finite dimensional vector space over and let
(a) Prove that ker ker if and only if for some

(b) Prove that there exists such that and

Please see the attached file for the fully formatted problems.

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Vector spaces, basis and quotient spaces are investigated in the solution. The solution is detailed and well presented.

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