Motivated by questions in theoretical computer science and quantum information theory, we study the classical problem of determining linear spaces of matrices of bounded rank. Spaces of bounded rank three were classified in 1983, and it has been a longstanding problem to classify spaces of bounded rank four. Before our study, no non-classical example of such a space was known. We exhibit two non-classical examples of such spaces and give the full classification of basic spaces of bounded rank four. There are exactly four such up to isomorphism. We also take steps to bring together the methods of the linear algebra community and the algebraic geometry community used to study spaces of bounded rank.

翻译：受理论计算机科学与量子信息理论问题的启发，我们研究经典问题：确定有界秩矩阵的线性空间。秩三的有界空间已于1983年被分类，而秩四的有界空间分类则是一个长期未解的难题。在此研究之前，尚未发现此类空间的非经典实例。我们给出了两个非经典实例，并完成了秩四基本空间的完全分类。在同构意义下，恰好存在四种这样的空间。此外，我们尝试融合线性代数领域与代数几何领域研究有界秩空间的方法，以促进学科间的协同。