Linear Mappings of Free Algebra: First Steps in Noncommutative Linear Algebra

For arbitrary universal algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in universal algebra, and the product of matrices is determined by the product of mappings. The system of equations, whose matrix is a matrix of mappings, is called a system of additive equations. I considered the methods of solving system of additive equations. As an example, I consider the solution of a system of linear equations over the complex field provided that the equations contain unknown quantities and their conjugates. Linear mappings of algebra over a commutative ring preserve the operation of addition in algebra and the product of elements of the algebra by elements of the ring. The representation of tensor product Aotimes A in algebra A generates the set of linear transformations of algebra A. The results of this research will be useful for mathematicians and physicists who deal with different algebras.