 The real matrices are in one-one correspondence with the linear mappings of the two-dimensional Cartesian coordinate system into itself by the rule

The next section displays M(2,R) is a union of planar cross sections that include a real line.M(2,R) is ring isomorphic to split-quaternions, where there is a similar union but with index sets that are hyperboloids.Within M(2, R), the multiples by real numbers of the identity matrix may be considered a real line.