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Problem (a) Let p : V V be a projection. Prove that V is the direct sum V = Kernel(p) + Image(p). (b) Suppose that V is a direct sum V = Prove that there is a projection p : V and Image(p) = V1. V such that Kernel(p) = V0 Problem distinct (a) Prove that if {L1, L2, L3} and (L'1, L'2, L'3) are two sets of pairwise disjoint one-dimensional subspaces of K2 (lines thru the origin), then there is a linear automorphism f : K² K2 such that, for all i = 1,2,3, f(Li) = L (b) Is the statement in (a) true for triples of pairwise distinct one-dimensional subspaces of K3? (c) Find two sets {P1, P2, P3} and (P', P'2,P's) of pairwise distinct two-dimensional subspaces of K3 (planes thru the origin) for which there is no linear automorphism f : K3 K3 such that, for all i = 1,2,3, f(Pi) = P.

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