. Proof, It follows that f ? (T ) = g(T ) ? f (T ) = f (T ) ? g(T ) since f T (T ) = 0 by definition of the minimal polynomial. Also, since f ? divides f , there is k ? R[x] so that f = kf ? and thus f, For the statement about images we have f (T ) = f ? (T ) ? k(T ) =? ?(f (T )) ? ?(f ? (T ))

. Lemma-8, Let v ? R n , then f v = f

. Proof, The first statement follows as if f (T ) annihilates v then it also annihilates g(T )(v) for any g ? R[x]. The second statement follows since, by the first isomorphism theorem, we have

. Lemma-9, Let v ? R n and f, g ? R[x] with f g = f v . Then f w = f for w = g(T )(v)

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