Problem 13. For each case, decide whether or not S is a subring of R.
(1) S is the set of all rational numbers of the form (a)/(b), where b is not divisible by 3 , and R=Q.
(2) S is the set of all functions which are linear combinations of the functions {1,cosnt,sinnt|ninZ}, and R is the set of all functions that map R->R.
(3) S is the set of real matrices of the form ([a,b],[-b,a]), and R is the set of all real 2 imes 2 matrices.
Problem 13. For each case, decide whether or not S is a subring of R. (1) S is the set of all rational numbers of the form a/b; where b is not divisible by 3, and R=Q.
(2) S is the set of all functions which are linear combinations of the functions {1,cos nt,sin nt |n E Z}, and R is the set of all functions that map R - R.
(3) S is the set of real matrices of the form
and R is the set of all real 2 x 2
matrices.