Definition of sum of linear transformations

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

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 \(\mathbb{R}^2\).Copy