  These pedagogical lecture notes address to the students in theoretical
physics for helping them to understand the mechanisms of the linear operators
defined on finite-dimensional vector spaces equipped with definite or
indefinite inner products. The importance of the Dirac conjugation is pointed
out presenting its general theory and a version of the Riesz theorem for the
indefinite inner product spaces, based on the Dirac-Riesz map that combines the
action of the Riesz map with that of the metric operator. The matrix
representations of the linear operators on vector spaces with definite or
indefinite inner products is also presented.
