Title: Linearly Mapping from Image to Text Space. Abstract: The extent to which text-only language models (LMs)  learn to represent the physical, non-linguistic world is an open question. Prior work has shown that pretrained LMs can be taught to ``understand'' visual inputs when the models' parameters are updated on image captioning tasks. We test a stronger hypothesis: that the conceptual representations learned by text-only models are functionally equivalent (up to a linear transformation) to those learned by models trained on vision tasks. Specifically, we show that the image representations from vision models can be transferred as continuous prompts to frozen LMs by training only a single linear projection. Using these to prompt the LM achieves competitive performance on captioning and visual question answering tasks compared to models that tune both the image encoder and text decoder (such as the MAGMA model). We compare three image encoders with increasing amounts of linguistic supervision seen during pretraining: BEIT (no linguistic information), NF-ResNET (lexical category information), and CLIP (full natural language descriptions). We find that all three encoders perform equally well at transferring visual property information to the language model (e.g., whether an animal is large or small), but that image encoders pretrained with linguistic supervision more saliently encode category information (e.g., distinguishing hippo vs.\ elephant) and thus perform significantly better on benchmark language-and-vision tasks. Our results indicate that LMs encode conceptual information structurally similarly to vision-based models, even those that are solely trained on images.