Igor wrote in [comment #5](https://forum.azimuthproject.org/discussion/comment/20805/#Comment_20805):

> In the definition of tensor product in Wikipedia it is unclear what do they mean by \\(e_i \otimes f_j\\), where \\(e_i\\) and \\(f_j\\) are basis vectors. They define \\(\otimes\\) for vector spaces, while not specifying what this means for individual vectors in these spaces.

They actually do explain this in the section [tensor product of vector spaces](https://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces): they say that if you choose a basis \\(e_i\\) of the vector space \\(V\\) and a basis \\(f_j\\) of the vector space \\(W\\), the vector space \\(V \otimes W\\) has a basis consisting of meaningless formal symbols \\(e_i \otimes f_j\\).

They don't actually say "meaningless formal symbols", but they say something that means the same if you're a mathematician: they say "each basis element is denoted \\(e_i \otimes f_j\\)". The word "denoted" here means "don't ask what it means: it's just a meaningless formal symbol".

There are dozens of ways to describe tensor products of vector spaces, and Wikipedia describes a few, but this is the quickest. Not the best, but the quickest.

> In the definition of tensor product in Wikipedia it is unclear what do they mean by \\(e_i \otimes f_j\\), where \\(e_i\\) and \\(f_j\\) are basis vectors. They define \\(\otimes\\) for vector spaces, while not specifying what this means for individual vectors in these spaces.

They actually do explain this in the section [tensor product of vector spaces](https://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces): they say that if you choose a basis \\(e_i\\) of the vector space \\(V\\) and a basis \\(f_j\\) of the vector space \\(W\\), the vector space \\(V \otimes W\\) has a basis consisting of meaningless formal symbols \\(e_i \otimes f_j\\).

They don't actually say "meaningless formal symbols", but they say something that means the same if you're a mathematician: they say "each basis element is denoted \\(e_i \otimes f_j\\)". The word "denoted" here means "don't ask what it means: it's just a meaningless formal symbol".

There are dozens of ways to describe tensor products of vector spaces, and Wikipedia describes a few, but this is the quickest. Not the best, but the quickest.