In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The proof solves linear systems of equations using that the Vandermonde matrices have always non zero determinants, when the entries of the power series of the rows of the matrix are all different. We provide several examples, and we also use our method to observe that the sequence of polynomials that we construct algebraically approaches the Taylor series of a function $f$ which is infinitely differentiable.