72. Transformation Sum

For linear transformations , define the sum pointwise:

The sum is itself linear.

72.1. Matrix correspondence

If has matrix and has matrix (so and ), then has matrix :

Let and (column-wise).

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)