Please see the attached file for the fully formatted problems.
Let P3 = ( it is set of all polynomials with coefficients in Z that are at most of degree 3.)
Let A = and B = where , that is  = .
(a) Verify that A and B are bases of the Z-module P3.
(c) Let D: P3 -> P3 be differentiation, i.e., ; e.g. .
Compute A and B . (Matrices of D with respect to A and B ).
Acting on an arbitrary linear combination of the basis elements in A:
and in :
(d) Let : P3  P3 be defined by
Verify that  is a Z - module homomorphism and compute A and B
Please see the attached file for the complete solution.
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(a) Since for each , we have . Then ...
Z-Modules of Polynomials, Basis and Linear Combinations are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.