Ordinary matrices can be thought of as the coordinate representations of linear maps between vector spaces (or free modules). Likewise, supermatrices can be thought of as the coordinate representations of linear maps between super vector spaces (or free supermodules). There is an important difference in the graded case, however. A homomorphism from one super vector space to another is, by definition, one that preserves the grading (i.e. maps even elements to even elements and odd elements to odd elements). The coordinate representation of such a transformation is always an "even" supermatrix. Odd supermatrices correspond to linear transformations that reverse the grading. General supermatrices represent an arbitrary ungraded linear transformation. Such transformations are still important in the graded case, although less so than the graded (even) transformations.