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# Z-Modules of Polynomials, Basis and Linear Combinations

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 &#61506; = .
(a) Verify that A and B are bases of the Z-module P3.

(b) Compute the change of basis matrices PAB (from the basis A to the basis B) and PB A (from the basis B to the basis A).

(c) Let D: P3 -> P3 be differentiation, i.e., ; e.g. .
Compute A and B . (Matrices of D with respect to A and B ).

Solution:
Acting on an arbitrary linear combination of the basis elements in A:
= DA
and in :

= DB

(d) Let &#61693;: P3 &#61614; P3 be defined by

Verify that &#61693; is a Z - module homomorphism and compute A and B

#### Solution Summary

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.

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