Existing vector space models typically map synonyms and antonyms to similar word vectors, and thus fail to represent antonymy. We introduce a new vector space representation where antonyms lie on opposite sides of a sphere: in the word vector space, synonyms have cosine similarities close to one, while antonyms are close to minus one. We derive this representation with the aid of a thesaurus and latent semantic analysis (LSA). Each entry in the thesaurus – a word sense along with its synonyms and antonyms – is treated as a “document,” and the resulting document collection is subjected to LSA. The key contribution of this work is to show how to assign signs to the entries in the co-occurrence matrix on which LSA operates, so as to induce a subspace with the desired property. We evaluate this procedure with the Graduate Record Examination questions of (Mohammed et al., 2008) and find that the method improves on the results of that study. Further improvements result from refining the subspace representation with discriminative training, and augmenting the training data with general newspaper text. Altogether, we improve on the best previous results by 11 points absolute in F measure.