Exercise 3.A.8

Question

Give an example of a function $\varphi:\mathbf{R}^2	o \mathbf{R}$ such that $$f(av)=af(v)$$for all $a\in \mathbf{R}$ and for all $v\in \mathbf{R}^2$ but $f$ is not linear.

Moon Flower · Accepted Answer

Consider a function $\varphi(v)=\sqrt[3]{v_{1}^3+v_{2}^3}$, where $v_{1}$ and $v_{2}$ are the first and second entries of $v$, respectively. We can verify that $$\varphi(av)=\sqrt[3]{(av_{1})^3+(av_{2})^3}=\sqrt[3]{a^3(v_{1}^3+v_{2}^3)}=a\sqrt[3]{v_{1}^3+v_{2}^3}=a\varphi(v).$$Additionally, since addition does not distribute over exponentiation, we can see that $\varphi(u+w)=\sqrt[3]{(u_{1}+w_{1})^3+(u_{2}+w_{2})^3}$ is not always equal to $\varphi(u)+\varphi(w)=\sqrt[3]{u_{1}^3+u_{2}^3}+\sqrt[3]{w_{1}^3+w_{2}^3}$; one such counter example is $u=\begin{bmatrix}1\0\end{bmatrix}$ and $w=\begin{bmatrix}0\1\end{bmatrix}$.

Exercise 3.A.8

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Exercise 3.A.8

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