Teaching the mathematics of quantum entanglement using elementary mathematical tools

We address the problem of teaching the mathematics of entanglement using only elementary linear algebra. For this goal, we first discuss tensor products using only matrix multiplication and with this we discuss entanglement for pure bipartite systems of arbitrary dimensions. We show how to assess entanglement using only Gaussian methods, i.e. the row reduced echelon form of the familiar Gauss-Jordan algorithm for solving systems of linear equations. In this way we can present entanglement avoiding the difficulties of tensor products and without the Schmidt decomposition. Some elementary examples are provided together with MATLAB scripts. A Gaussian algorithm for the factorization of unentangled states is given.