Definition of a tensor field Could anybody explain to me the following:

If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$
  where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative,
  then $T_{ij}$ is a tensor field.

I am new to all hese covector, covariant, etc. and am very confused.
I believe the definition of a tensor is one which transforms according to the tensor transformation rule... right? But I don't see how that applies here. 
Thank you in advance.

Alternatively to Neal, you can approach this in a coordinate-free way using the Jacobian and some wedge products.  
In particular, if $f(x) = x'$ defines a coordinate transformation, then $\underline f(a) = a \cdot \partial f$ is the Jacobian, and $\overline f(a)$ denotes the transpose.  Covectors transform according to the transpose.  That is,  
$$V = \overline f(V'), \quad \nabla = \overline f(\nabla')$$  
where the primes refer to objects in the new coordinates.  If $T$ is a tensor, it must follow $T = \overline f(T')$.  We know that $T = \nabla \wedge V$, so $T' = \nabla' \wedge V'$ if $T$ is a tensor.  Let's check that:  
$$T = \nabla \wedge V = \overline f(\nabla') \wedge \overline f(V') = \overline f(\nabla ' \wedge V') = \overline f(T')$$  
If you can convert that approach to index notation, you deserve credit for the problem.
