Orthogonal Matrices: Properties and Applications

This lecture covers the properties and applications of orthogonal matrices, focusing on the concept of orthogonality in linear algebra. Starting with the definition of an orthogonal family of vectors, the instructor explains how to determine if a family is orthogonal or orthonormal. The lecture then delves into the importance of orthogonal bases and how to decompose vectors in these bases. Additionally, the lecture explores the concept of orthonormal matrices and their significance in linear transformations. Theorem 6.18 is introduced to establish the conditions for a matrix to be orthonormal. The lecture concludes with a discussion on orthogonal projections and their applications in vector spaces.