Quote by mathwonk
tensors are combinations of vectors and functions on vectors. contracting means you have one of each type and you evaluate the function on the vector (or multivector).

You mean linear functions on vectors  and those I prefer to call forms.
And please don't talk about contraction before you define what a tensor is... Because if you do (and people do) than everything related to tensors shrinks to manupulation of components...
And still I find your description somehow inaccurate: you should specify what does your vectors mean in every single place in your statement. (e.g. [tex]\alpha \in V^*[/tex] is oneform on [tex]V[/tex] but vector in [tex]V^*[/tex])