reference frames - How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having difficulty understanding this kind of definition. To me, it goes like this:

1. 
   

   First we have a collection of bases or coordinate systems (do these represent reference frames?) and the transformations between them.

   
   


2. 
   

   A vector/tensor is an assignment of an array of numbers to each basis, and these arrays are related to each other by coordinate transformations.

   
   

I am wondering how coordinate systems and transformations are specified.

1. 
   
   

   Do we need all possible coordinate systems and transformations to define a vector space or just a few of them?

   
   


2. 
   

   How to define the concept of basis and coordinate transformation without completing the very notion of vector?