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Determine the Eigenvalues of a Function

K is a field, V is the vector space of all polynomials over K, D is the derivative function from V to V and g is a linear function from V to V given by

g(f(x)) = xDf(x).

Find all eigenvalues of g. When are there infinitely many eigenvalues? Please explain in detail.

PLEASE NOTE: it states that g is a linear function from V to V. This means f lies in the domain of g i.e f lies in V. Therefore f is a polynomial and cannot be the exponential function

Solution Summary

Eigenvalues are found. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.