I am trying to understand the concept of tensors. I seem to understand that they are generalization of vectors: They are subject to similar basis transformations with vectors but I am somewhat confused about their relations with matrices. In 3 dimensional Euclidian space, a rank 1 tensor is represented with a vector, a rank 2 tensor is represented with a 3x3 matrix, a rank 3 tensor is a 3x3x3 sized three dimensional array and so on. Assuming we are in 3 dimensional space, for example, is it possible to have an arbitrary sized matrix; say 6x7 sized matrix as the representation of a tensor? Or can they be only represented with arrays with dimensions of three's powers? I think that this is not possible since an arbitrary sized matrix cannot undergo basis transformations.
My second question is about tensor multiplications. This operation seems like a cartesian product between the elements of two tensors, for example the tensor product of a rank 1 (3 elements) and rank 2 (9 elements) tensor generates a rank 3 three tensor (27 elements). Is this operation related to standard matrix multiplication somehow? For example, we can do matrix multiplication between 3x1 and 1x3 vectors and obtain a 3x3 matrix. Isn't this a tensor product between two rank 1 tensors in the same time? Is this a coincidence?