False.
The range of the transformation T is not necessarily equal to R^m.
The range of a linear transformation T: R^n -> R^m is the set of all possible output vectors of T, i.e., the set of all vectors y in R^m such that there exists an input vector x in R^n such that T(x) = y.
The range of a transformation T can be thought of as the span of the columns of the matrix A that represents T, which is the set of all possible linear combinations of the columns of A.
Therefore, the range of the transformation T will depend on the column space of A, which is a subspace of R^m, and not necessarily equal to R^m. The dimension of the column space of A will give the rank of the matrix A, and the rank of A can be at most min(m, n).