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Let T: V -> V be a linear map such that T2 - I = 0...

Let T: V -> V be a linear map such that T2 - I = 0 where I is the identity map on V.

a) Prove that Im(T-I) is a subset of Ker(T+I) and Im(T+I) is a subset of Ker(T-I).

b) Prove that V is the direct sum of Ker(T-I) and Ker(T+I).

c) Suppose that V is finite dimensional. True or false there exists a basis B of V such that [T]B is a diagonal matrix. Justify your answer.

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Answer #1

then B=B1 U B2 is the required basis in which matrix of T is Diagonal matrix

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