  Matrix product states play an important role in quantum information theory to
represent states of many-body systems. They can be seen as low-dimensional
subvarieties of a high-dimensional tensor space. In these notes, we consider
two variants: homogeneous matrix product states and uniform matrix product
states. Studying the linear spans of these varieties leads to a natural
connection with invariant theory of matrices. For homogeneous matrix product
states, a classical result on polynomial identities of matrices leads to a
formula for the dimension of the linear span, in the case of 2x2 matrices.
  These notes are based partially on a talk given by the author at the
University of Warsaw during the thematic semester "AGATES: Algebraic Geometry
with Applications to TEnsors and Secants", and partially on further research
done during the semester. This is still a preliminary version; an updated
version will be uploaded over the course of 2023.
