Today's physics students would profit by being introduced to Dirac's bra-ket notation prior to encountering it in the unfamiliar context of quantum mechanics. While the notation may fruitfully be used in dealing with dc circuit theory, vectors in three-space, and the inertial tensor; we here use it to facilitate the succinct presentation of the linear transformations which specify the relations among systems of units. With this theory the student may be introduced to bra-ket notation on familiar ground, gain some insight into the structure of systems of units, and acquire a tool for computing conversion factors between any two systems of units he may encounter or devise.