Definition. An application $f:U\rightarrow \mathbb{R}^{n}$ defined in the open $U\subset \mathbb{R}^{m}$ is differentiable at point $a\in U$ if there is a linear application $T:\mathbb{R}^{m}\rightarrow \mathbb{R}^{n}$ such that $f\left ( a+v \right )-f\left ( a \right )=T.v+r\left ( v \right )$,where $\lim_{v\rightarrow 0}\frac{r\left ( v \right )}{\left | v \right |}=0$.
This linear application $T$ is called the $derivative$ of $f$ at point $a$, and is denoted by ${f}'\left ( a \right )$.
We know that given the vector spaces $\mathbb{R}^{m},\mathbb{R}^{n}$ the application $T:\mathbb{R}^{m}\rightarrow \mathbb{R}^{n}$ is linear if:
1) $T\left ( v+w \right )=T\left ( v \right )+T\left ( w \right )$, $\forall v,w\in \mathbb{R}^{m}$
2) $T\left ( \alpha v \right )=\alpha T\left ( v \right )$, $\forall \alpha \in \mathbb{R}$, $\forall v\in \mathbb{R}^{m}$.
My question is that I can not visualize conditions 1) and 2) so $T$ is a linear application in the given definition.
Thanks for your help