In this paper, we develop a new approach to studying Appell polynomial sequences via linear algebra. This approach offers a powerful tool for investigating the properties of Appell polynomials. Within a self-contained theory, employing only matrix operations, we show how the derivation of some of old and new properties of Appell polynomials is greatly simplified.

Appell polynomials, Bernoulli polynomials and numbers, Euler polynomials and numbers, generalized Pascal functional matrix, Wronskian vector, generalized Appell identity.