Transformation Sum

72. Transformation Sum

For linear transformations $𝑇, 𝑆 : ℝ^{𝑛} \to ℝ^{𝑚}$ , define the sum $𝑇 + 𝑆 : ℝ^{𝑛} \to ℝ^{𝑚}$ pointwise:

(𝑇 + 𝑆) (\vec{𝑥}) = 𝑇 (\vec{𝑥}) + 𝑆 (\vec{𝑥})

The sum is itself linear.

72.1. Matrix correspondence

If $𝑇$ has matrix $𝐴$ and $𝑆$ has matrix $𝐵$ (so $𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥}$ and $𝑆 (\vec{𝑥}) = 𝐵 \vec{𝑥}$ ), then $𝑇 + 𝑆$ has matrix $𝐴 + 𝐵$ :

Let $𝐴 = [\begin{matrix} {\vec{𝑎}}_{1} & {\vec{𝑎}}_{2} & \dots & {\vec{𝑎}}_{𝑛} \end{matrix}]$ and $𝐵 = [\begin{matrix} {\vec{𝑏}}_{1} & {\vec{𝑏}}_{2} & \dots & {\vec{𝑏}}_{𝑛} \end{matrix}]$ (column-wise).

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

So in matrix terms: transformation addition = matrix addition, performed entrywise.

72.2. Connections

Scalar multiplication of transformations — companion operation
Linear Transformation — the underlying object
Matrix Multiplication — composition is the multiplicative analog (composition ↔ product, sum ↔ sum)