DSE 2012 option b question 13

[rarj144]

It would be great help if some one could solve this question.

thanksM.A.-Entrance-2012-Option-B.pdf

[Noel]

I was only able to do a part of propositions B and C..for B it was
rank(AA-)=dim(R(AA-))
=dim{AA-(x)|x€W}
=dim{A(v)|v€V}
=dim(R(A))
=rank(A)

i could not prove the other part of proposition B
and similarly you can prove nullity(AA-)=nullity(A) for proposition C
i'm sorry but i could only do this trivial bit