Relational style laws and constructs of linear algebra

We present a few laws of linear algebra inspired by laws of relation algebra. The linear algebra laws are obtained from the relational ones by replacing union, intersection, composition and converse by the linear algebra operators of addition, Hadamard product, composition and transposition. Many of the modified expressions hold directly or with minor alterations. We also define operators that sum up the content of rows and columns. These share many properties with the relational domain and codomain operators returning a subidentity corresponding to the domain and codomain of a relation. Finally, we use the linear algebra operators to write axioms defining direct sums and direct products and we show that there are other solutions in addition to the traditional —in the relational context— injection and projection relations.