We focus on a modular, graphical language—graphical linear algebra—and use it as high-level language for calculational reasoning. We propose a minimal framework of axioms that highlight the dualities and symmetries of linear algebra, and showcase the resulting diagrammatic calculus. Our work develops a relational approach to linear algebra, closely connected to classical relational algebra. With the symmetrical high-level axioms we are able to provide a fully diagrammatic proof that a fragment of Graphical Linear Algebra is equivalent to matrix algebra.