Definition of sum of linear transformations

Definition: If (S, T: V o W) are linear transformations, their sum (S + T) is defined pointwise:
[(S + T)(\mathbf{v}) = S(\mathbf{v}) + T(\mathbf{v}) \quad ext{for all } \mathbf{v} \in V]

The sum is computed by applying both transformations to the same input and adding the results in the codomain (W).

Example: If (T(x,y) = (x, 0)) and (S(x,y) = (0, y)), then ((T+S)(x,y) = (x, y)), which is the identity transformation on (\mathbb{R}^2).