01 | Dual space , dual basis , transformation of bases and components

This is the first lecture of a course on mathematical physics for physicists, focusing on differential geometry and Lie groups. The instructor begins with organizational details and introduces the course structure, emphasizing that the first weeks will cover linear algebra, which is foundational for differential geometry. He quotes Raoul Bott, stating that 80% of mathematics is linear algebra, to highlight its importance. The lecture then delves into the concept of a vector space, its dimension, and the notion of basis, stressing that all bases are equally valid without additional structure like a scalar product. The change-of-basis matrix is introduced, and its properties, such as invertibility, are discussed. The main topic is the dual space L*, defined as the space of linear maps from L to R. The instructor constructs the dual basis and proves that the dual space has the same dimension as the original space. He also discusses the bilinear canonical pairing between a vector space and its dual. The lecture concludes with the transformation rules for components of vectors and co-vectors under a change of basis, emphasizing the distinction between contravariant and covariant components. The presentation is rigorous, with detailed explanations and examples, suitable for physics students.