![[sol-5.pdf#page=26]] --- $\textbf{Exercise 12.}$ Suppose $S, T \in \mathcal{L}(V, W)$. (a) Prove that $(S + T)^* = S^* + T^*$. (b) Prove that $(\lambda T)^* = \lambda T^*$ for all $\lambda \in \mathbb{F}$. This exercise asks you to verify (a) and (b) in 3.120. $\textbf{Solution 12.}$ (a) We have $\begin{align*} (S + T)^*(\varphi) &= \varphi \circ (S + T) \\ &= \varphi \circ S + \varphi \circ T \\ &= S^*(\varphi) + T^*(\varphi). \end{align*}$ Thus $(S + T)^* = S^* + T^*$, as desired. (b) Suppose $\lambda \in \mathbb{F}$. Then $\begin{align*} (\lambda T)^*(\varphi) &= \varphi \circ (\lambda T) \\ &= \lambda(\varphi \circ T) \\ &= \lambda T^*(\varphi). \end{align*}$ Thus $(\lambda T)^* = \lambda T^*$, as desired. --- Problem 12 (Section 3F): $\textbf{Exercise 12.}$ Suppose $S, T \in \mathcal{L}(V, W)$. (a) Prove that $(S + T)^* = S^* + T^*$. (b) Prove that $(\lambda T)^* = \lambda T^*$ for all $\lambda \in \mathbb{F}$. This exercise asks you to verify (a) and (b) in 3.120. $\textbf{Solution 12.}$ (a) We have $\begin{align*} (S + T)^*(\varphi) &= \varphi \circ (S + T) \\ &= \varphi \circ S + \varphi \circ T \\ &= S^*(\varphi) + T^*(\varphi). \end{align*}$ Thus $(S + T)^* = S^* + T^*$, as desired. (b) Suppose $\lambda \in \mathbb{F}$. Then $\begin{align*} (\lambda T)^*(\varphi) &= \varphi \circ (\lambda T) \\ &= \lambda(\varphi \circ T) \\ &= \lambda T^*(\varphi). \end{align*}$ Thus $(\lambda T)^* = \lambda T^*$, as desired. $\textit{Commentary:}$ This exercise verifies two basic properties of the dual map: it preserves addition and scalar multiplication. These properties show that the dual map is a linear map between the space of linear maps from $V$ to $W$ and the space of linear maps from $W^*$ to $V^*$. The proofs are straightforward calculations using the definition of the dual map and the properties of composition and scalar multiplication of linear maps. These properties are crucial for understanding the functorial nature of the dual operation and its role in category theory. $\textit{Example:}$ Let $V = \mathbb{R}^2$, $W = \mathbb{R}^3$, and define $S, T \in \mathcal{L}(V, W)$ by $\begin{align*} S(x, y) &= (x, y, 0), \\ T(x, y) &= (0, x, y). \end{align*}$ For $\varphi = (a, b, c) \in W^* = (\mathbb{R}^3)^*$, we have $\begin{align*} S^*(\varphi)(x, y) &= \varphi(S(x, y)) = ax + by, \\ T^*(\varphi)(x, y) &= \varphi(T(x, y)) = bx + cy. \end{align*}$ Thus, for $\lambda \in \mathbb{R}$, $\begin{align*} (S + T)^*(\varphi)(x, y) &= \varphi((S + T)(x, y)) = ax + 2bx + cy \\ &= S^*(\varphi)(x, y) + T^*(\varphi)(x, y), \\ (\lambda T)^*(\varphi)(x, y) &= \varphi((\lambda T)(x, y)) = \lambda bx + \lambda cy \\ &= \lambda T^*(\varphi)(x, y). \end{align*}$ ---