--- Commentary: $\textbf{Exercise 8.}$ Give an example of a function $\varphi : \mathbb{R}^2 \to \mathbb{R}$ such that $\varphi(av) = a\varphi(v)$ for all $a \in \mathbb{R}$ and all $v \in \mathbb{R}^2$ but $\varphi$ is not linear. $\textit{Commentary:}$ This exercise and the next exercise show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map. $\textbf{Solution 8.}$ Define $\varphi : \mathbb{R}^2 \to \mathbb{R}$ by $\varphi(x, y) = (x^3 + y^3)^{1/3}.$ Then $\varphi(av) = a\varphi(v)$ for all $a \in \mathbb{R}$ and all $v \in \mathbb{R}^2$. However, $\varphi$ is not linear, because $\varphi(1, 0) = 1$ and $\varphi(0, 1) = 1$ but \begin{align*} \varphi((1, 0) + (0, 1)) &= \varphi(1, 1) \ &= 2^{1/3} \ &\neq \varphi(1, 0) + \varphi(0, 1). \end{align*} Of course there are also many other examples. $\textit{Commentary:}$ This exercise shows that homogeneity alone (i.e., the property that $\varphi(av) = a\varphi(v)$ for all $a \in \mathbb{R}$ and all $v \in \mathbb{R}^2$) is not enough to ensure that a function $\varphi : \mathbb{R}^2 \to \mathbb{R}$ is linear. The given function $\varphi(x, y) = (x^3 + y^3)^{1/3}$ satisfies homogeneity, but fails to satisfy additivity, as shown by the example $\varphi(1, 0) + \varphi(0, 1) \neq \varphi((1, 0) + (0, 1))$. Geometrically, a function that satisfies homogeneity but not additivity can distort the shape of the input space in a way that is inconsistent with linear transformations. For example, the function in this exercise maps the unit square to a unit cube, which is not a linear transformation. Additivity alone is also not enough to imply that a function is a linear map, although the proof of this involves advanced tools that are beyond the scope of this book. $\textit{Examples:}$ The function $\varphi : \mathbb{R}^2 \to \mathbb{R}$ defined by $\varphi(x, y) = |x| + |y|$ satisfies homogeneity but not additivity, and hence is not linear. The function $\varphi : \mathbb{R}^2 \to \mathbb{R}$ defined by $\varphi(x, y) = \max(x, y)$ satisfies homogeneity but not additivity, and hence is not linear. The function $\varphi : \mathbb{C} \to \mathbb{C}$ defined by $\varphi(z) = |z|$ (the modulus of $z$) satisfies homogeneity but not additivity over $\mathbb{C}$, and hence is not $\mathbb{C}$-linear. These examples illustrate other functions that satisfy homogeneity but not additivity, including functions involving absolute values and maximum values. They demonstrate that homogeneity is a weaker condition than linearity, and that functions satisfying homogeneity can still have significant nonlinear behavior.