A large part of computing deals with the doing, undoing, and reinterpretation of linear transformations on vector spaces. These operations are essential to the description of most physical phenomena, are a core part of machine learning, and also form the basic building blocks of quantum algorithms. The tendency of all things to evolve towards thermal equilibrium poses challenges for efficient and accurate computation, which often appear in the form of energy dissipation and errors caused by thermal noise. While thermal physics has provided some insights about physical limitations on computing, less is known about the particular considerations that apply to linear algebraic computations. Here, I study the application of thermodynamics to such problems, in both classical and quantum systems. Several new algorithms are given for linear algebra problems using specialized hardware, which have theoretical advantages over previously existing algorithms. Additionally, new bounds are proven for the energy cost of unitary transformations of quantum states, as well as for the efficiency of communication between quantum systems.