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