Visualizing $T = T^{\mu}_\nu \frac{\delta}{\delta x^{\mu} } \otimes dx^{\nu} $ If $ T $ is a $(1,1)$ tensor on a manifold, and
$$T = T^{\mu}_\nu \frac{\delta}{\delta x^{\mu} } \otimes dx^{\nu} $$ .
how do I visualize this, in order to prove something with it? I need to prove 
 $$  F_{*}(T^{\mu}_\nu \frac{\delta}{\delta x^{\mu} } \otimes dx^{\nu} ) = T^{\mu}_\nu \frac{\delta y^{\alpha}}{\delta x^{\mu} }\frac{\delta y^{\nu}}{\delta x^{\beta} }\frac{\delta }{\delta y^{\alpha} }\otimes \delta y^{\beta} $$,
but, even though I have already discussed this problem with other people, I haven't understood the problem, I beleive it is because I can't visualize this, as to get an idea of what to prove, and the starting point of that is how to visualize something like T
Any comment, response would be much appreciated
 A: $\newcommand{\dd}{\partial}$The tensor
$$
T = T_{\nu}^{\mu}\, \frac{\dd}{\dd x^{\mu}} \otimes dx^{\nu}
$$
is the endomorphism field with matrix $T_{\nu}^{\mu}$ with respect to the coordinate frame $(\frac{\dd}{\dd x_{\nu}})$. If $F$ is a local diffeomorphism (so that $F$ pushes forward $1$-forms via $F_{*} dx^{\nu} = (F^{-1})^{*} dx^{\nu}$), the transformation law
$$
F_{*}(T_{\nu}^{\mu} \frac{\dd}{\dd x^{\mu}} \otimes dx^{\nu})
= T_{\nu}^{\mu} \frac{\dd y^{\alpha}}{\dd x^{\mu}} \frac{\dd x^{\nu}}{\dd y^{\beta}}\, \frac{\dd}{\dd y^{\alpha}} \otimes dy^{\beta}
$$
follows from the chain rule in the form
$$
F_{*} dx^{\nu} = (F^{-1})^{*} dx^{\nu} = \frac{\dd x^{\nu}}{\dd y^{\beta}}\, dy^{\beta},
\qquad
F_{*} \frac{\dd}{\dd x^{\mu}} = \frac{\dd y^{\alpha}}{\dd x^{\mu}}\, \frac{\dd}{\dd y^{\alpha}}.
$$
Conceptually, this transformation rule may be viewed as analogous to the similarity $A' = PAP^{-1}$ if $A$ and $A'$ are matrices of a single linear transformation with respect to bases $B$ and $B'$, and $P$ is the transition matrix from $B$ to $B'$. (Here, $(P^{-1})_{\beta}^{\nu} = \frac{\dd x^{\nu}}{\dd y^{\beta}}$ and $P_{\mu}^{\alpha} = \frac{\dd y^{\alpha}}{\dd x^{\mu}}$.)
