Question about statement of the exercise 8.3.1(d) from Zorich

Question

In this exercise the author defines two paths $x_{1}$ and $x_{2}$ to be equivalent at the point $x_{0}\in \mathbb{R}^m$ if $x_{1}(0)=x_{2}(0)=x_{0}$ and $d(x_{1}(t),x_{2}(t))=o(t)$ as $t\to 0$ (I assume it is meant that $x_{1}$ and $x_{2}$ are defined in some neigborhood of $0$). Part (a) of the exercise asks to show that it is indeed an equivalence relatio. Part (b) asks to show that there is a one-to-one correspondence between vectors from $T_{x_{0}}\mathbb{R}^m $ (which at this point in the book is just defined as a copy of $\mathbb{R}^m$ attached at the point $x_{0}$) and equivalence classes of paths smooth at $x_{0}$. Part (c) asks to use this correspondence to define addition and scalar multiplication for these equivalence classes. Now finally, part (d) asks to check whether these just defined operations depend on the coordinate system used in $\mathbb{R}^m$. I had no problem understanding statements of the first three parts and doing them, but I don't think I properly understand what I am asked to check in part (d). I assume that by different coordinate systems in $\mathbb{R}^m$ author means different bases of $\mathbb{R}^m$ but I still don't see what exactly I am asked to prove or find a counterexample to. So what is the precise statement that I need t check in this exercise ?