Paul Garret's Proof of the Cayley-Hamilton Theorem I am trying to understand the proof of Cayley-Hamilton Theorem given in Paul Garrett's notes.
We have a finite dimensional vector space $V$ over a field $k$ and are given a linear operator $T\in \mathcal L(V)$.
The proof on pg 431 in the above link starts out as:
The module $V\otimes_k k[x]$ is free of rank $\dim_k(V)$ over $k[x]$. Also, $V$ is also a $k[T]$ module by the action $v\mapsto Tv$. (I understand this much).
Now the next line reads:
So $V\otimes_k k[x]$ is a $k[T]\otimes_k k[x]$ module.

This I do not understand.

What is the general fact at play here?
EDIT: To expand on my (complete) lack of understanding, I do not see how are we giving a $k[T]\otimes_k k[x]$ module structure to $V\otimes_k k[x]$.
And of course, I am looking for a 'general principle' at work here.
For example, when we said that $V\otimes_k k[x]$ is a $k[x]$-module of rank $\dim_k(V)$, what we are using is the following: We have a natural injection $i:k\to k[x]$. So we can extend the scalars on the $k$-module $V$ and get a $k[x]$-module $V\otimes_k k[x]$.
Since $V\otimes_k k[x]\cong k[x]^n$, where $n=\dim_k(V)$, we also know that the rank of $V\otimes_k k[x]$ as a $k[x]$-module is same as $\dim_k(V)$.

$V$ is a $k[T]$-module.  
$V \otimes $ Something is a ($k[T] \otimes$ Something) - module.  
