Let T be the linear operator on R2 the matrix of which in the standard ordered b

Question

Let T be the linear operator on R2 the matrix of which in the standard ordered basis is A = Prove that the only subspaces of R2 invariant under T are R2 and the zero subspace. WE HAVE T[U] =T[X,Y]'=A [X,Y]' = ZERO SUBSPACE AND R 2 ARE ALWAYS INVARIANT IN 2 D UNDER THE MATRIX A GIVEN.. WE NEED ONLY TO PROVE THAT THERE ARE NO ONE DIMENSIONAL INVARIANT SUB SPACES SO LET US CHECK THEM EIGEN VALUES OF A ARE GIVEN BY SO THETRE NO EIGEN VALUES IN THE REAL FIELD HENCE THERE ARE NO REAL INVARIANT SUBSPACES IN 1 D SO WE HAVE THE ONLY INVARIANT SPACES AS THOSE IN 0 D ZERO SUB SPACE AND ..2D .THAT IS THE R2 SUB SPACE

Explanation / Answer

your proof is correct

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