> (individual numbers (0 dimensions), vectors (1 dimension), matrixes (2 dimensions), and so on) You're using the word "dimension" in two (distinct) ways. Inst

> (individual numbers (0 dimensions), vectors (1 dimension), matrixes (2 dimensions), and so on) You're using the word "dimension" in two (distinct) ways. Instead, use the word "rank": > (individual numbers (0 rank), vectors (1 rank), matrixes (2 rank), and so on) Now, we can talk about a 4-dimensional rank-1 tensor, e.g., a 4-element vector. Now, think about a 4x4 matrix: if we multiply the matrix by a 4-vector, we get a 4-vector out: in some ways, the multiplication has "eaten" one of the ranks of the matrix; but, the dimension of the resulting object is the same. If we had a 3x2 matrix, and we multiplied it by a 3-vector, then both the rank has changed (from 2 to 1) and the dimension has changed (from 3 to 2). A tensor has any number of rank. More importantly, the ranks of a tensor come in two "flavors": a vector and a one-form. The concepts are pretty darn general, but one way to get a feel for how they're related is that the transpose of a vector can be its dual. This gets into things like pre- and post- multiplication; or, whether we 'covary' or 'contravary' with respect to the tensor. Frankly, tensor products are a beast to deal with, mechanically, so the literature mostly deals with them as opaque objects. The modern tensor software libraries and high performance computing has seen a sea-change in the use of GR.