Great vid explaining the essence of a tensor -- matching up components with basis vectors.
Rank 0 (scalar) : component, but no basis vectors Rank 1 (vector) : component & 1 basis vector (1 x 3) Rank 2 (array) : component matched with 2 basis vectors. (3x3)
(1 basis vector could be direction of surface, the other is the force applied in the x-direction).
Key useful property (which I need to understand): the combination of components & basis vectors can be agreed-upon by all reference frames (moving, rotating, etc.). What does the "combination" refer to? ("Coordinate-free" -- express relationship between vectors). More on coordinate-free geometry: http://www.cs.brown.edu/~ls/teaching08/LN04_Coordfreegeom.pdf
No absolute coordinates -- you just have a "point" and a vector (with direction and magnitude). But it's not on some grid with x and y components. Coordinates are needed to implement, but not reason about, the coordinate system.