  The purpose of this paper is to articulate a coherent and easy-to-understand
way of doing quantum mechanics in any finite-dimensional Liouville space, based
on the use of Kronecker product and what we have termed the `bra-flipper'
operator. One of the greater strengths of the formalism expatiated on here is
the striking similarities it bears with Dirac's bra-ket notation. For the
purpose of illustrating how the formalism can be effectively employed, we use
it to solve a quantum optical master equation for a two-level quantum system
and find its Kraus operator sum representation. The paper is addressed to
students and researchers with some basic knowledge of linear algebra who want
to acquire a deeper understanding of the Liouville space formalism. The
concepts are conveyed so as to make the application of the formalism to more
complex problems in quantum physics straightforward and unencumbered.
