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Eigenvector/Dual space/Dual basis/Exercise

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Let

be an endomorphism on a finite-dimensional -vector space , and let be an eigenvector for with eigenvalue . Let

be the dual mapping of . We consider bases of of the form with the dual basis . Give examples of the following behavior.

a) is an eigenvector of with the eigenvalue independent of .


b) is an eigenvector of with the eigenvalue with respect to some basis , but not with respect to another basis .


c) is for no basis an eigenvector of .