The Change of Basis Groupoid

Change of basis in finite-dimensional vector spaces has numerous significant applications. This research explores the algebraic structure of change of basis matrices within a set of \(m\) bases of a finite-dimensional vector space using category theory. The investigation reveals a connected groupoid of order \(m^2\) whose morphisms correspond to change of basis matrices. Subgroupoids within this structure correspond to upper and lower triangular matrices and matrices with alternating elements of \(0\). We identify bases leading to triangular change of basis matrices. For univariate polynomial families, they occur where the minimum degree of the polynomials increases, or the maximum degree decreases. Similarly, we find that alternating bases lead to alternating change of basis matrices. These bases occur with basis polynomials that have definite parity such as Chebyshev polynomials of the first kind. A commutative diagram elucidates the subgroupoids with morphisms corresponding to triangular and alternating change of basis matrices.