Conclusion
I think it is not exactly correct that the normal transform is an inverse transpose. A normal vector is not a column vector, the transform is a left inverse as Equation 5 shows. The transpose comes from the implementation of the vector class. This should be a left inverse as a rigorous formulation. I don't say the references [2,3,6] are incorrect, but I prefer the references [4,5] since they distinguish these two kind of vectors.

However, the references [2,3,6] have an easier to understand explanations. Especially, [2,3,6] shows why normal transformation doesn't work with applying

However, the references [2,3,6] have an easier to understand explanations. Especially, [2,3,6] shows why normal transformation doesn't work with applying

*A*intuitively. I like these explanations more. Here I would like to highlight the difference of a usual vector and a normal vector, they are a column vector or a row vector. By the way, contravariant vector means doesn't-change-vector, but, its representation can change. It seems I should explain the difference of these vectors are related with inner product. I would like to study mo…