Definition of scalar multiple of a linear transformation

Definition: If (T: V o W) is a linear transformation and (c) is a scalar, the scalar multiple (cT) is defined pointwise:
[(cT)(\mathbf{v}) = c \cdot T(\mathbf{v}) \quad ext{for all } \mathbf{v} \in V]

The transformation (cT) scales every output of (T) by the factor (c).

Example: If (T(x,y) = (2x, 3y)) and (c = 5), then ((5T)(x,y) = (10x, 15y)).

Example: If (D) is the derivative operator on polynomials, then ((3D)(p) = 3p').