Linear Operators: Basis Transformation and Eigenvalues

This lecture covers the basis transformation of vectors in a vector space over a field, focusing on two bases and the invertible matrix characterizing the transformation. It also delves into the motivation for studying linear operators, particularly in inner product spaces, and the importance of eigenvalues in Quantum Mechanics. The lecture explores essential linear algebra concepts, including adjoint, self-adjoint, unitary, and normal operators, as well as eigenvectors, eigenspaces, and the Spectral Theorem for normal operators.