Eigenvalues and Eigenvectors: Definitions, Examples

This lecture covers the definitions and examples of eigenvalues and eigenvectors in linear algebra, focusing on transformations of vector spaces and diagonalization. It explains how to identify eigenvalues and corresponding eigenvectors, illustrating with practical examples such as orthogonal projections. The lecture also discusses the simplicity of diagonal matrices for manipulation and the properties of eigenvalues and eigenvectors in matrix transformations. Additionally, it explores the concept of null vectors and the injectivity of transformations, emphasizing the importance of non-trivial solutions. Overall, the lecture provides a comprehensive understanding of eigenvalues and eigenvectors in the context of linear transformations.