Singular Values and Norms: Understanding Linear Maps with SVD

This lecture delves into the concept of singular value decomposition (SVD) of a matrix A, explaining how it always exists and its significance in linear algebra. Through the SVD, the instructor demonstrates how applying a linear map to a vector affects its norm, showcasing the properties of orthogonal matrices and their impact on vector norms. By analyzing the relationship between singular values and vector norms, the lecture concludes with a fundamental result in linear algebra, illustrating the essential role of SVD in understanding linear maps.