$f^*\left(\sum_I a_I dx_I\right) = \sum_I (f^*a_I) df_I$

$f^*\left(\sum_I a_I dx_I\right) = \sum_I (f^*a_I) df_I$

By linearity of $f^*$, $$f^*\left(\sum_I a_I dx_I\right) = \sum_I (f^*
a_I) dx_I$$
And if I want $df_I$, I would have to do $$f^*\left(\sum_I a_I dx_I\right)
= \sum_I a_I (f^*dx_I) = \sum_I a_I df_I.$$
But the book gives $$f^*\left(\sum_I a_I dx_I\right) = \sum_I (f^*a_I)
df_I.$$
Can anyone demystify it for me?