Wronskian theory revisited from a linear algebraic and null space viewpoint

This paper aims at providing a new fundamental framework for the Wronskian theory by paying attention to the null space of the Wronski matrix. It is first shown that identical vanishing of the Wronskian leads to two different representations of vector-valued functions that lie in the null space of the Wronski matrix. It is further shown that they are algebraically linearly dependent, by which we are immediately led to a key identity in the Wronskian theory, derived only through very simple linear algebraic arguments. Most fundamental results on the Wronskian theory available in the literature can thus be obtained in a quite straightforward fashion with a very clear perspective, and hence the relevant arguments are believed to be of pedagogical value, too. Some further issues relevant to the null space viewpoint are also discussed, where some other results available in the literature are derived through the null space viewpoint in a somewhat strengthened form.