Is there relationship between tensor as an operator and its contraction?

Question

According to Wikipedia article on Tensor.

The space of type (1, 1) tensors $V \otimes V^*$ is isomorphic in a natural way to the space of linear transformations from V to V.

Is there relation between such and the contraction of $V \otimes V^*$?

Answer 1

Given an element $v\otimes f$ of $V\otimes V^*$ you can construct:

1) Its contraction $f(v)$, which produce a scalar, in $\mathbb R$, if $V$ is a vector space over this field, since $f:V\to\mathbb R$ is linear functional (by the definition of $V^*$).

2) A linear transformation $V\to V$ through $w\mapsto f(w)v$. This assignment, $V\otimes V^*\to\hom(V,V)$, is an isomorphism between vector spaces.

Answer 2

If you have a linear map $m:V\to V$ and you use this isomorphism to treat it as an element of $V\otimes V^*$ then you can apply contraction to get an element of the underlying field. This is precisely the trace of $m$, $\mathrm{tr}(m)$.