We give a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard bases is constructed. For some special cases of this matrix, matrix equations including inverse matrices related the Bell polynomials and Diophantine equation are obtained. With the help of these equations, new formulas containing different polynomials with the Bernoulli polynomials are found. In order to compute these polynomials, a computational algorithm is given. Finally, by applying the Laplace transform to the generating function for the Bernoulli polynomials, we derive some novel formulas involving the Hurwitz zeta function and infinite series.