Define $[n] = \{1,\ldots, n\} $, where $n \in \mathbb{N}$ and define the $2^n$- dimensional $\mathbb{R}$-algebra $C_n$ as follows:
Notation: Basis is $e_I$, where $I \subset \mathbb{N}$ and let $e_i$ = $e_I$ when $I = \{i\}$.
Multiplication in $C_n$ is defined by the rules $e_\emptyset = 1$, $e _ie_j = -e_je_i$ for $i \neq j \in [n]$, $e_i^2 = 1$ for $i \in [n]$, and $e_I = e_{i_1}, \ldots, e_{i_k}$ where $I=\{i_1, \ldots,i_k\}$ with $i_1 < \ldots < i_k \in [n]$
Question: Right multipilcation by $x \in C_n$ is a $\mathbb{R}$-linear endomorphism, show that:
$$tr(e_I) = \begin{cases} 2^n &\mbox{if } I = \emptyset\\ 0 & \mbox{if } I \neq \emptyset \end{cases} $$