Definition of scalar multiple of a linear transformation
Created almost 9 years ago, updated 5 months ago
ID24017
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').
**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 \((3D)(p) = 3p'\).