If we need to construct the tensor product of two operators, and we do already know the matrix elements of the building blocks, we can combine them directly.
Here is the rule for combining by example 2x2 matrices to form 4x4 matrices:
Remark 1: the same pattern works for matrices of any size. In general, the product of an m x n matrix and a p x q matrix is a mp x nq matrix.
Remark 2: All of this applies perfectly well to column and rows vectors, which are just some kind of matrices.
so we would write by example for the basis vector |uu> of a qubit system: