A map is linear if the following is given:
I. $f(\lambda*\vec{x})=\lambda f(\vec{x})$
II. $f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})+f(\vec{x_2})$
Now I am able to show that a map is linear on basic maps, but I have run into a problem where I am unsure.
$\varphi: \mathbb{R}^n \mapsto \mathbb{R}, \begin{pmatrix} x_{1} \\ \vdots \\ x_{n} \end{pmatrix} \mapsto \sum\limits_{k=1}^n|x_k|$
What I have tried to this point:
I. $\varphi(\lambda* \vec{x}) = \sum\limits_{k=1}^n|\lambda *x_k| \neq \sum\limits_{k=1}^n \lambda|x_k| = \lambda*\varphi(\vec{x})$
II.
Now about the second one I am not sure anymore. The problem I see is that I can not apply the $\varphi(\vec{x_1}+\vec{x_2})=f(\vec{x_1})+f(\vec{x_2})$ since the sum has a $x_k$ in it.
$\varphi$ is not linear; consider the case where $\vec{x_1} = -\vec{x_2}$.