Transformation Scalar Multiplication

73. Transformation Scalar Multiplication

For a linear transformation $𝑇 : ℝ^{𝑛} \to ℝ^{𝑚}$ and a scalar $𝑐 \in ℝ$ , define the scaled transformation $𝑐 𝑇 : ℝ^{𝑛} \to ℝ^{𝑚}$ pointwise:

(𝑐 𝑇) (\vec{𝑥}) = 𝑐 𝑇 (\vec{𝑥})

The scaled transformation is itself linear.

73.1. Matrix correspondence

If $𝑇$ has matrix $𝐴 = [\begin{matrix} {\vec{𝑎}}_{1} & {\vec{𝑎}}_{2} & \dots & {\vec{𝑎}}_{𝑛} \end{matrix}]$ , then $𝑐 𝑇$ has matrix $𝑐 𝐴$ :

\begin{aligned} (𝑐 𝑇) (\vec{𝑥}) & = 𝑐 (𝑇 (\vec{𝑥})) \\ = 𝑐 (\sum_{𝑖 = 1}^{𝑛} 𝑥_{𝑖} {\vec{𝑎}}_{𝑖}) \\ = \sum_{𝑖 = 1}^{𝑛} 𝑥_{𝑖} (𝑐 {\vec{𝑎}}_{𝑖}) \\ = (𝑐 𝐴) \vec{𝑥} \end{aligned}

So scaling a transformation = scaling its matrix entrywise.

73.2. Combined with sum: vector space of transformations

Together with sum, scalar multiplication makes the set of linear transformations $ℝ^{𝑛} \to ℝ^{𝑚}$ into a vector space of dimension $𝑚 𝑛$ — isomorphic to the space of $𝑚 \times 𝑛$ matrices.

73.3. Connections

Transformation Sum — companion operation
Vector Operations — same structure (scaling distributes over sum) at the vector level
Linear Transformation — the underlying object