We adapt methods coming from additive combinatorics to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Kneser's theorem on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras without using Galois correspondence arguments.