  Much of the knowledge encoded in transformer language models (LMs) may be
expressed in terms of relations: relations between words and their synonyms,
entities and their attributes, etc. We show that, for a subset of relations,
this computation is well-approximated by a single linear transformation on the
subject representation. Linear relation representations may be obtained by
constructing a first-order approximation to the LM from a single prompt, and
they exist for a variety of factual, commonsense, and linguistic relations.
However, we also identify many cases in which LM predictions capture relational
knowledge accurately, but this knowledge is not linearly encoded in their
representations. Our results thus reveal a simple, interpretable, but
heterogeneously deployed knowledge representation strategy in transformer LMs.
