Finitedimensional linear transformations are used a lot in theoretical physics. They form a group with respect to multiplication, and some of these groups are extremely useful. So we are dealing with matrix products. In theoretical physics, you'll sooner or later encounter some of the following notation for multiplication groups of matrices. n always stands for dimension:

GL(n,C) complex matrices

GL(n,R) real matrices

SL(n,C) complex matrices with unit determinant

U(n) unitary matrices

SL(n,R) real matrices with unit determinant

SU(n,R) unitary matrices with unit determinant

O(n) orthogonal (unitary and real) matrices

SO(n) orthogonal with unit determinant
Examples:
 The above groups are all subgroups of GL(n,C).
 SL(n,R), O(n) and SO(n) are subgroups of GL(n,R).
 SO(n) is a subgroup of O(n), same with SU(n) and U(n)
 SO(3) is the group of rotations in 3dimensional space.
 U(1) is the gauge group of quantum electrodynamics
 SU(3) is the gauge group of quantum chromodynamics
 SU(2) x U(1) is the gauge group of electroweak theory