Definition: If \(T: V \to W\) is a linear transformation and \(c\) is a scalar, the scalar multiple \(cT\) is defined pointwise:
\[(cT)(\mathbf{v}) = c \cdot T(\mathbf{v}) \quad \text{for all } \mathbf{v} \in V\]
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 \((3D)(p) = 3p'\).
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