Why a vector is a (1,0) tensor?

Question

I am looking for some familiar examples of Tensors and I am wondering why a vector is a (1,0) tensor type?

That is it takes some covector and gives and scalar!! How?

Answer 1

For a finite-dimensional vector space $V$, $V$ is isomorphic to its double dual $V^{**},$ through the natural isomorphism $\varphi:V\rightarrow V^{**}$ given by $\varphi(x)(v)=v(x),$ where $x\in V$ and $v\in V^*.$ Hence, an element of $V$ can be seen as a linear functional that acts on covectors.