In A Geometric Approach to Differential Forms [Bachman, David], the author states the following:
(On the geometric interpretation of 1-forms in $\mathbb{R^2}$) - 'We can interpret the act of multiplying by a constant geometrically. Suppose $\mathbb{w}$ is given by $adx+bdy$. Then the value of $\mathbb{w}$(V1) is the length of the projection of V1 onto the line, $l$, where $\frac{<a,b>}{|<a,b>|^2}$ is a basis vector for $l$. This interpretation has a huge advantage... it is coordinate free. Forms are objects that exist independently of our choice of coordinates.'
I understand the geometric intuition, however, I fail to see why this is coordinate free. I would like to find out where I go wrong. What I see is that, as $\mathbb{w}$ relies on $dx$ and $dy$, the differentials of coordinate functions themselves, how can it then be coordinate free? Furthermore, in $\frac{<a,b>}{|<a,b>|^2}$, $<a.b>$ and thus its magnitude also depends on $dx$ and $dy$ so I fail to see how this is independent of coordinates - rather, the opposite as it makes an explicit reference to the differentials of coordinate functions?
Would be very grateful if someone can show me at what point my logic goes faulty...
Note: The author defines $dx$ and $dy$ as the coordinate function for $T_p\mathbb{R^2}$, such that a vector in $T_p\mathbb{R^2}$ can be written $<dx,dy>$ where $<1,0>=\frac{d(x+t,y)}{dt}$ and $<0,1>=\frac{d(x,y+t)}{dt}$