(1) Show that the eigenvalues of a matrix are an invariant, that is, that they depend only on the linear function the matrix represents and not on the choice of basis vectors.

(2) Show also that the eigenvectors of a matrix are not an invariant.

(3)Explain why the dependence of the eigenvectors on the particular basis is exactly what we would expect and argue that is some sense they are indeed invariant.

Please show detailed explanation and provide an example for each question.