QR Factorization: Orthogonal Bases and Matrices

This lecture covers the QR factorization theorem, stating that a matrix A with linearly independent columns can be decomposed as A=QR, where Q is an orthogonal matrix with orthonormal columns and R is an upper triangular invertible matrix. The Gram-Schmidt orthogonalization process is used to obtain an orthonormal basis of the column space of A. Examples illustrate the QR factorization process and the construction of orthogonal bases. The lecture also introduces the concept of orthogonal matrices and their properties, emphasizing their importance in numerical computations and solving systems of equations.