Intuitively, a tensor is an object which has two attributes:
- invariant direction under a change of coordinates, and
- components which change in a ”special” way under a change of coordinates.
The idea is, that if you have some forward and backwards transformations be-
tween two coordinate basis, those transformations are inverses of one another
and behave consistently. In a more mathematically rigorous sense, a tensor
is defined as some set of vectors and covectors combined together under a
tensor product, hence the name ”tensor”.