Linear maps using Tensor Product While I was reading some posts (Definition of a tensor for a manifold, and Tensors as matrices vs. Tensors as multi-linear maps), I encountered the following explanation:
"To give a linear map $V \rightarrow V$ is the same as to give a linear map $V^* \otimes V\rightarrow \mathbb{R}$, assuming we're looking at real vector spaces."
Could anybody kindly explain the above sentense in detail with an example?   I am not a math-major, but very much interested in tensor analysis.  Thank you in advance. 

Hint:
In slightly more generality, which actually makes it a bit easier to tell what's going on, suppose that $V$ and $W$ are finite dimensional $\Bbb{R}$-vector spaces.
The map
$$
\begin{align}
V^* \otimes W &\to \hom(V, W) \\
f \otimes w &\mapsto \bigg( v \mapsto f(v)w \bigg)
\end{align}
$$
is a natural isomorphism.  (It doesn't depend on choice of bases.)
We can precompose with other isomorphisms, but the duality map in the $\color{red}{\text{middle}}$ is not natural so we have to pick a basis and corresponding dual basis.
$$
\left( V^* \otimes W \right)^* = \hom(V^* \otimes W, \Bbb{R}) \color{red}{\to} V^* \otimes W \to \hom(V, W)
$$
