# Eigenvalues

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? Give an example of a 5x5 matrix A over the real numbers such that A has precisely three eigenvalues, 0,1 and 2 and the image of the linear transformation has dimension three. Justify your answer.

? Let V be the set of real valued sequences . Check that V is a vector space over the field of real numbers, using addition of vectors + = and scalar multiplication = . Let T be the shift map sending to with . Check that T is a linear transformation and compute its eigenvalues and eigenvectors.

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#### Solution Summary

This shows how to construct a matrix with three eigenvalues, check that a set is a vector space, and check that T is a lineear transformation, computing eigenvalues and eigenvectors.

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