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1. Assume that V is a vector space and L is a linear function V →...

1. Assume that V is a vector space and L is a linear function V → V.

a. Suppose there are two vectors v and w in V such that v, w, and v+w are all eigenvectors of L. Show that v and w share the same eigenvalue.

b. Suppose that every vector in V is an eigenvector of L. Prove that there is a scalar α such that L = αI.

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