Sumários
Multi-linear algebra - III
12 Março 2024, 10:00 • Giordano Cotti
This lecture included remarks and comments on previous lectures. We clarified the distinction between decomposable and non-decomposable tensors with explicit examples. Special attention was given to tensors defined by bilinear forms on the product of two vector spaces, introducing the concept of a non-degenerate bilinear form. It was shown that such forms (not necessarily symmetric) establish a canonical (left/right) identification between a vector space and its dual. Additionally, it was noted that a non-degenerate bilinear form in a vector space of dimension >1 does not yield a decomposable (0,2)-tensor.
Multi-linear algebra - II
6 Março 2024, 15:30 • Giordano Cotti
In this lecture, I introduced the concept of the "universal problem" for tensor products, which aims to linearize multilinear maps of vector spaces universally. I discussed the existence and uniqueness of solutions to this problem. Firstly, I demonstrated that if a solution exists, it is unique up to a unique isomorphism. Secondly, I explicitly constructed the solution to the universal problem as a space of multilinear forms. Additionally, I proved that the tensor product of finite-dimensional vector spaces is also finite-dimensional, and I provided constructions of bases composed of decomposable tensors. Finally, I showed that the tensor product of vector spaces is associative up to canonical isomorphisms.
Multi-linear algebra - I
5 Março 2024, 10:00 • Giordano Cotti
In this lecture we recalled basic results in linear algebra, and introduced basics notions in multi-linear algebra.