So you have to show that:
T(S) is a subset of R^m (show that T(s) is in R^m for each s in S)
If u and v are in T(S), u+v is in T(S)
If u is in T(S) and c is any scalar, cu is in T(S).
Using the fact that T is linear and that S is a subspace of R^n, these should be fairly straightforward to prove.
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