My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category.
I know that any biproduct category admits a calculus of matrix. What I'm wondering is whether we can describe a linear transformation through the individual action of each element of the corresponding matrix. For example, would it be possible to describe an element of $GL(2,\mathbb{R})$ with this diagram, wherein the objects are 1-dimensional vector spaces ?
I've tried to see if the usual matrix multiplication relation holds, but this would require adding more relations, and it seems to lead to contradictions. Has this been studied before ?