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Definition: A transformation T: R^n -> R^m is called a linear transformation if 1) T(u + v) = T(u) + T(v) for all u and v in R^n and 2) T(cv) = cT(v) for all v in R^n and all scalars c.
Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)