The matrix of the quadratic forms giving the squared moduli of the lattice vectors can be taken as a representation of the lattice. The formalism adopted is capable of direct connection between real and reciprocal space and allows one to assume the canonical definition of reduced form as a definition of reduced cell. The theory of congruent transformations of the quadratic forms can be applied to the solution of a variety of crystallographic problems as axes and indices transformations, unit cell reduction and can help in understanding the relationships between chemically similar but non-isomorphous solids.