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Let A be element of Endomorphism of V (V is a finite dimensional vector space over F) such that A is onto. Assume that there exist a function B: V $\to$ V such that BA = I. Prove that AB = I

  • Can you give me a hint on how to prove this problem? Thanks.

Here is working solution. Since A is onto, there exist x in V such that A(x) = v.

We need to show that BA = I.

(BA)(x) = B(A(x)) = B(v) then I don't know what's next

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    Is $V$ a vector space? Finite-dimensional?2012-10-31
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    Yes, it is a finite dimensional vector space.2012-10-31
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    There is no need for $V$ being a vector space. This is valid for surjective functions $f: X \to Y$.2012-10-31
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    See http://math.stackexchange.com/questions/3852/if-ab-i-then-ba-i, http://math.stackexchange.com/questions/216569/assuming-ab-i-prove-ba-i, etc.2012-11-01

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