@zacattack In some sense, yes! Though I would actually argue with you a bit here. To know how a tensor behaves, you also have to know how many times covariant/contravariant it is. Otherwise, you don't know whether to apply it to vectors or covectors. So if I just hand you a matrix, I haven't really given you a full description of a tensor. (I also need to tell you what basis I'm using to encode the tensor.) So, as with linear maps, matrices (and multidimensional matrices) are merely a representation of tensors, but not the full story.
This is essentially a tensor of rank 2
@zacattack In some sense, yes! Though I would actually argue with you a bit here. To know how a tensor behaves, you also have to know how many times covariant/contravariant it is. Otherwise, you don't know whether to apply it to vectors or covectors. So if I just hand you a matrix, I haven't really given you a full description of a tensor. (I also need to tell you what basis I'm using to encode the tensor.) So, as with linear maps, matrices (and multidimensional matrices) are merely a representation of tensors, but not the full story.