I am trying to understand a line of reasoning in Guillemin and Pollack. Let $U\subset \mathbb{R}^k$ and $V\subset \mathbb{R}^l$ be open sets, and let $f:V\rightarrow U$ be smooth. Use $x_1,...,x_k$ for the standard coordinate functions on $\mathbb{R}^k$ and $y_1,...,y_l$ on $\mathbb{R}^l$.
I am trying to understand the following equation: $f^*dx_i=\sum_{j=1}^l\frac{\partial f_i}{\partial y_j}dy_j=df_i$, where $f^*$ denotes the pullback by $f$.
For a vector $v\in V$, we have $f^*dx_i(v)=dx_i(df(v))$, but I'm not sure where to go from here. I know that $df$ pushes the vector forward into $\mathbb{R^k}$, and that $dx_i$ in turn measures the $i$th coordinate of that vector; however, I am confused why it should be equal to the some on the right. How are they equal?