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Recent results have established necessary and sufficient conditions for a nonlinear system of the form . with , to be locally equivalent in a neighborhood of the origin in Rnto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that span an -dimensional space and that is an involutive set.