Silviu said:
Hello! I am reading about tensors and I am a bit confused about rank-2 tensors. From what I understand they can be represented by a matrix.
Yes. This assumes, however, the choice of some basis, according to which the matrix entries are the coordinates.
However I am not sure I understand the difference between (2,0), (0,2) and (1,1) tensors.
It's the same as between ##f : U^* \times V^* \rightarrow \mathbb{F}\, , \,f : U \times V \rightarrow \mathbb{F}\, , \,f: U^* \times V \rightarrow \mathbb{F}##.
I understand that they act on different objects (vectors or one forms or both) ...
Yes.
... but having a matrix, ...
requires a basis of both ...
how can u know what kind of tensor it is?
You can't. How can you tell, whether ##(1,2)## is a vector, a linear function ##f : \mathbb{R}^2 \rightarrow \mathbb{R}\, , \,x \mapsto \langle (1,2),x\rangle## or simply a point in the Euclidean plane? Or a lattice point?
Also, for example I saw definitions of electromagnetic field strength tensor written both as (2,0) and (0,2) tensor and I am not sure I understand the difference.
As ##V^* \cong V## the difference is, whether ##(1,2) \in V## or ##(x \mapsto \langle (1,2),x \rangle) \in V^*##. Both are represented by ##(1,2)##.
What would be the 2 vectors it can act on and what would be the 2 one-forms it can act on, and how do we know when to use the right form?
It simply depends on what you want the matrix ##M## to represent. You have ##f(u,v)= u^tMv## and from which vector spaces ##u## and ##v## are, depends on what you want to do.