Orthogonal Bases in Vector Spaces

This lecture covers the concept of orthogonal bases in vector spaces, where a set of vectors is considered an orthogonal base if it is both orthogonal and unitary. The lecture explains how any element in the vector space can be uniquely expressed in terms of an orthogonal base. It also delves into the notion of orthonormal bases, which are orthogonal bases with unit vectors. The lecture provides examples and proofs related to orthogonal bases, projections, and spectral decomposition. Additionally, it discusses the spectral decomposition theorem, which states that any vector in a vector space can be uniquely represented as a sum of vectors from orthogonal subspaces.