The sum of linear transformations is a linear transformation

**Theorem:** If \(S, T: V \to W\) are linear transformations, then \(S + T\) is also a linear transformation.

**Proof:**
1. *Additivity:* \((S+T)(\mathbf{u} + \mathbf{v}) = S(\mathbf{u} + \mathbf{v}) + T(\mathbf{u} + \mathbf{v}) = S(\mathbf{u}) + S(\mathbf{v}) + T(\mathbf{u}) + T(\mathbf{v}) = (S(\mathbf{u}) + T(\mathbf{u})) + (S(\mathbf{v}) + T(\mathbf{v})) = (S+T)(\mathbf{u}) + (S+T)(\mathbf{v})\)

2. *Homogeneity:* \((S+T)(c\mathbf{v}) = S(c\mathbf{v}) + T(c\mathbf{v}) = cS(\mathbf{v}) + cT(\mathbf{v}) = c(S(\mathbf{v}) + T(\mathbf{v})) = c(S+T)(\mathbf{v})\)

Both linearity conditions are satisfied, so \(S + T\) is linear.

**Intuition:** The pointwise sum of two structure-preserving maps is itself structure-preserving because addition in the codomain is compatible with the linearity of each map.Copy