--- Commentary: $\textbf{Exercise 9.}$ Give an example of a function $\varphi : \mathbb{C} \to \mathbb{C}$ such that $\varphi(w + z) = \varphi(w) + \varphi(z)$ for all $w, z \in \mathbb{C}$ but $\varphi$ is not linear. (Here $\mathbb{C}$ is thought of as a complex vector space.) $\textit{Commentary:}$ There also exists a function $\varphi : \mathbb{R} \to \mathbb{R}$ such that $\varphi$ satisfies the additivity condition above but $\varphi$ is not linear. However, showing the existence of such a function involves considerably more advanced tools. $\textbf{Solution 9.}$ Define $\varphi : \mathbb{C} \to \mathbb{C}$ by $\varphi(x + yi) = x - iy$ for all $x, y \in \mathbb{R}$. Then $\varphi$ is additive but $\varphi$ is not homogeneous with respect to complex scalars. For example,$ \begin{align*} \varphi(ii) &= \varphi(-1) \\ &= -1 \\ &\neq 1 \\ &= i(-i) \\ &= i\varphi(i) \end{align*}$ and thus $\varphi(ii) \neq i\varphi(i)$. $\textit{Commentary:}$ This exercise shows that additivity alone (i.e., the property that $\varphi(w + z) = \varphi(w) + \varphi(z)$ for all $w, z \in \mathbb{C}$) is not enough to ensure that a function $\varphi : \mathbb{C} \to \mathbb{C}$ is linear over $\mathbb{C}$. The given function $\varphi(x + yi) = x - iy$ satisfies additivity, but fails to satisfy homogeneity over $\mathbb{C}$, as shown by the example $\varphi(ii) \neq i\varphi(i)$. Geometrically, the function in this exercise is a reflection across the real axis in the complex plane. This is an additive operation (reflecting the sum of two numbers is the same as summing their reflections), but it is not a $\mathbb{C}$-linear operation, because it does not commute with multiplication by complex scalars. There also exist functions $\varphi : \mathbb{R} \to \mathbb{R}$ that satisfy additivity but not homogeneity over $\mathbb{R}$, and hence are not $\mathbb{R}$-linear. However, constructing such functions is much more difficult and requires tools from advanced real analysis, such as non-measurable sets or the Axiom of Choice. $\textit{Examples:}$ 1. The function $\varphi : \mathbb{C} \to \mathbb{C}$ defined by $\varphi(x + yi) = x + 2yi$ satisfies additivity but not homogeneity over $\mathbb{C}$, and hence is not $\mathbb{C}$-linear. 2. The function $\varphi : \mathbb{C} \to \mathbb{C}$ defined by $\varphi(x + yi) = y + xi$ (rotation by 90° counterclockwise) satisfies additivity but not homogeneity over $\mathbb{C}$, and hence is not $\mathbb{C}$-linear. 3. The function $\varphi : \mathbb{C} \to \mathbb{C}$ defined by $\varphi(z) = \bar{z}$ (complex conjugation) satisfies additivity but not homogeneity over $\mathbb{C}$, and hence is not $\mathbb{C}$-linear. However, it is $\mathbb{R}$-linear when $\mathbb{C}$ is viewed as a vector space over $\mathbb{R}$. These examples illustrate other functions that satisfy additivity but not homogeneity over $\mathbb{C}$, including functions that involve changing the imaginary part of a complex number or applying complex conjugation. They demonstrate that additivity is a weaker condition than $\mathbb{C}$-linearity, and that functions satisfying additivity can still have significant nonlinear behavior over $\mathbb{C}$.