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For a class of nonlinear systems, we investigate the problem of under what conditions there exists a coordinate transformation that yields a state affine linear system up to output injection with implicit outputs. In particular, we provide necessary and sufficient conditions for time-varying linearization up to multi-output injection. We highlight that if the conditions hold, as a consequence, it is possible in the new coordinates to construct an observer with linear error dynamics. We propose a methodology to find the coordinate transformation. Several examples illustrate the proposed procedure.