Let $V$ be vector space, $\dim V=N$. Define the multiplication operator $L_{\mathbf{b}}$ as $L_{\mathbf{b}}:\omega\to \mathbf{b}\wedge\omega$, where $\omega\in\wedge V$ ($\wedge V$ is the entire exterior algebra) and $\mathbf{b}\in V$. I want to compute the trace of $L_{\mathbf{b}}$.
If $N=1,2$ choosing a basis in $V$ we can obtain a basis of $\wedge V$ and with some calculations we obtain $tr(L_{\mathbf{b}})=0$. However this method will be very complicated for larger value of $N$ because $\dim(\wedge V)=2^N$. So I search for a coordinate-free (basis-free) calculations applying the following definition of trace.
If $A=\sum_{k=1}^N \mathbf{v}_k\otimes\mathbf{f}_k^{*}\in V\otimes V^{*}$ then $tr(A)=\sum_{k=1}^N \mathbf{f}_k^{*}(\mathbf{v}_k)$.