How is the product $c_I \alpha^I$ defined?

Question

My Doubt:-

The author had pointed out the notations used by him. I have put that in a box. In the Proposition 3.29, the author used the notation for proving linear independence using $\sum c_{I}\alpha^{I}$. According to the paragraph in the box $c_I=(c_{i_1},...,c_{i_k})$ and $\alpha^I=\alpha_{i_1}\wedge ... \wedge \alpha_{i_k}$. On contradictory to the notation mentioned in the box auther had written $C_I \in \mathbb R$. How does the product $c_I \alpha^I$ defined? From Diego's answer. I have one more doubt. How can I expand the summation? For example, if the summation is like $\sum_{k=1}^n a_ic^i$. Then, without any confusion, I can expand it as $a_1c^1+...+a_nc^n$. Please help me.

Answer 1

Here $c_I$ isn't a tuple, but a number associated with the multi-index $I$. He specifically says so at "$c_I \in \mathbb{R}$".