I'm pretty sure this is the first time I've ever tried to answer another person's questions about anything tensor-related, so I apologize if I get something (or everything) totally wrong.

@Igor I imagine the wikipedia article is saying that \\(e_i\\) and \\(f_j\\) are basis vectors to two vectors spaces \\(E\\) and \\(F\\), which you then want to tensor together to get \\(E \otimes F\\). It sounds like you're then asking how to find the basis vectors of \\(E \otimes F\\).

Suppose both \\(E\\) and \\(F\\) are two-dimensional; I'll call their basis vectors \\(e_1, e_2, f_1, f_2\\).

To write down all the \\(e_i \otimes f_j\\), you just combine every way you can (that fits the pattern):

\\(e_1 \otimes f_1\\)

\\(e_1 \otimes f_2\\)

\\(e_2 \otimes f_1\\)

\\(e_2 \otimes f_2\\)

Those are your four basis vectors of \\(E \otimes F\\). (Recall that we expected 4, because the dimension of \\(E \otimes F\\), which is itself a vector space, is dim(E) * dim(F) = 2*2 = 4.)

@Igor I imagine the wikipedia article is saying that \\(e_i\\) and \\(f_j\\) are basis vectors to two vectors spaces \\(E\\) and \\(F\\), which you then want to tensor together to get \\(E \otimes F\\). It sounds like you're then asking how to find the basis vectors of \\(E \otimes F\\).

Suppose both \\(E\\) and \\(F\\) are two-dimensional; I'll call their basis vectors \\(e_1, e_2, f_1, f_2\\).

To write down all the \\(e_i \otimes f_j\\), you just combine every way you can (that fits the pattern):

\\(e_1 \otimes f_1\\)

\\(e_1 \otimes f_2\\)

\\(e_2 \otimes f_1\\)

\\(e_2 \otimes f_2\\)

Those are your four basis vectors of \\(E \otimes F\\). (Recall that we expected 4, because the dimension of \\(E \otimes F\\), which is itself a vector space, is dim(E) * dim(F) = 2*2 = 4.)