#math/linear-algebra Suppose $T\in \mathcal{L}(V,W)$. Then 1. $(S+T)'=S'+T'$ for all $S\in \mathcal{L}(V,W)$; 2. $(\lambda T)'=\lambda T'$ for all $\lambda \in \mathbf{F}$; 3. $(ST)'=T'S'$ for all $S\in \mathcal{L}(W,U)$. **1 Proof:** Suppose $T\in \mathcal{L}(V,W)$. Let $S\in \mathcal{L}(V,W)$ be an arbitrary linear map. By definition, $T'(\varphi)=\varphi \circ T:V\to \mathbf{F}$ with $\varphi \in \mathcal{L}(W,\mathbf{F})$. Additionally, the function $(S+T)'$ is defined by $(S+T)'=\varphi \circ(S+T):V\to \mathbf{F}$. Suppose an arbitrary $v\in V$ and $\varphi \in W'$, then $ \begin{align} ((S+T)'\varphi)(v) & = (\varphi \circ(S+T))(v) & \text{By def of dual maps} \\ & =\varphi((S+T)v) & \text{By def of func compositions} \\ & =\varphi(Sv+Tv) & \text{Addition of linear maps}\\ & =\varphi(Sv)+\varphi(Tv) & \text{Additivity of linear maps}\\ & = (S'\varphi)v + (T'\varphi)v & \text{By def of dual maps} \end{align} $ **2 Proof:**