  While an explicit basis is common in the study of Euclidean spaces, it is
usually implied in the study of inertial relativistic systems. There are some
conceptual advantages to including the basis in the study of special
relativistic systems. A Minkowski metric implies a non-orthonormal basis, and
to deal with this complexity the concepts of reciprocal basis and the vector
dual are introduced. It is shown how the reciprocal basis is related to upper
and lower index coordinate extraction, the metric tensor, change of basis,
projections in non-orthonormal bases, and finally the Gram-Schmidt procedure.
It will be shown that Lorentz transformations can be viewed as change of basis
operations. The Lorentz boost in one spatial dimension will be derived using
the Gram-Schmidt orthonormalization algorithm, and it will be shown how other
Lorentz transformations can be derived using the Gram-Schmidt procedure.
