What is really meant with the word "generator" in Bott & Tu? How can a form "generate" a whole vector space of other forms?

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    $\begingroup$ Same as in generators of a group, or generators of a module. Each de Rham cohomology group is a vector space over $\Bbb R$, after all. $\endgroup$ – Lord Shark the Unknown Aug 12 '18 at 15:11

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