TITLE: An basic question of linear algebra and the vectorial spaces.
QUESTION [1 upvotes]: I'm studying Differencial Geometry for a exam and I'm having some trouble to do this. In this course we viewed dual spaces, forms, Gauss Bonnet Theorem... Hm.. this is it!

Let $V$ a vectorial space, $\dim(V)=n$ and $T:V \rightarrow V$ an linear aplication. The trace of $T$ is the number $tr(T)$ such that: 
  $$
\sum_{i=1}^n\Delta(v_1,...,v_{i-1},T(v_i),...,v_n)=tr(T)\Delta(v_1,...,v_n)
$$
  where $\Delta \in \bigwedge^n(V^*)$ 
  If $(e^1,...,e^n)$ is a base of $V$ and $T(e_i)=\sum a_i^je_j$ show that 
$$
tr(T)=\sum a_i^i
$$

I have not idea how to do this, can someone help me?
Thank you!

REPLY [2 votes]: Use the basis vectors $e_i$ in place of the vectors $v_1, v_2, \ldots$ in the definition of the trace.  Because $\Delta$ is an $n$-form, it is alternating.  Any term of the form $\Delta(\ldots, e_j, \ldots, a_i^j e_j, \ldots)$ must evaluate to zero.
For this reason, the only terms that contribute are those of the form
$$\sum_{i=1}^n \Delta(e_1, e_2, \ldots, T(e_i), \ldots) = \sum_{i=1}^n \Delta(e_1, e_2, \ldots, a^i_i e_i, \ldots)$$
I think you should be able to take it from here?
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