A modal accessibility relation is just a transition relation, and so can be represented by a transition matrix. As a matter of fact, multimodal logics can also be represented by generalized transition matrices provided matrix entries are taken to be in a more complex Boolean algebra than the standard two-valued one. Starting from this point, I show that bisimulations have a rather elegant theory when expressed in terms of transformations on Boolean vector spaces. The resulting work is a curious hybrid, fitting between conventional modal semantics and conventional linear algebra. I don't know where the investigations begun here will ultimately wind up, but in the meantime the approach has a kind of curious charm that others may find appealing.