On the Kernel Method Compared to the Matrix Inversion Method for Passing From a Dual Basis to the Basis of a Vector Space

In this work, we study and compare two methods for passing from a dual basis to the basis of a finite-dimensional vector space by recalling the passage from a basis to its dual. For the inverse transition, we clarify the kernel method for linear forms and the matrix inversion method, the former exploiting the properties of linear forms and orthogonality, while the latter relies on the explicit inversion of transition matrices, with remarks for each approach depending on the context of application. The study shows that, although matrix inversion is a more direct method, the kernel method can offer a more elegant and efficient alternative in certain cases, especially when the vector space has particular structures.