Linear Mathematics | Part IB, 2002

Let α\alpha be an endomorphism of a vector space VV of finite dimension nn.

(a) What is the dimension of the vector space of linear endomorphisms of VV ? Show that there exists a non-trivial polynomial p(X)p(X) such that p(α)=0p(\alpha)=0. Define what is meant by the minimal polynomial mαm_{\alpha} of α\alpha.

(b) Show that the eigenvalues of α\alpha are precisely the roots of the minimal polynomial of α\alpha.

(c) Let WW be a subspace of VV such that α(W)W\alpha(W) \subseteq W and let β\beta be the restriction of α\alpha to WW. Show that mβm_{\beta} divides mαm_{\alpha}.

(d) Give an example of an endomorphism α\alpha and a subspace WW as in (c) not equal to VV for which mα=mβm_{\alpha}=m_{\beta}, and deg(mα)>1\operatorname{deg}\left(m_{\alpha}\right)>1.

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