In Bott&Tu's differential forms in algebraic topology he defines.
Let $\Omega^*$ be the algebra over $\Bbb R$ generated by $dx_1, \ldots, dx_n$ with the relations $$ \begin{cases} (dx_i)^2 & = 0 \\ dx_i dx_j &= -dx_j dx_i, \, i \not= j . \end{cases} $$
And define the differential form as elements of
$$ \Omega^*(\Bbb R^n) = C^\infty(\Bbb R^n) \otimes _{\Bbb R} \Omega^*.$$
Then the authors claim
If $w$ is such a form $w$ can be uniquely written as $\sum f_Id_I$. for some increasing index $I$.
Why unique? I see that $d_I$ forms a basis in $\Omega^*$ itself. How does one prove this? I am curious also what is the general approach in proving some sets forms a basis when given relations?
My thoughts:
Relation in tensor products always seem confusing. My final thought would be work with decomposition. (= is congruence here)
$$ C^\infty(\Bbb R^n) \otimes \Omega^* = C^\infty(\Bbb R^n) \otimes_{\Bbb R} \bigoplus_I \Bbb R_I = \bigoplus_{I} C^{\infty} \otimes_{\Bbb R} \Bbb R_I = \bigoplus_I C^\infty(\Bbb R^n)_I $$