Lecture
This lecture introduces the concept of orthogonal bases in vector spaces, defining orthonormal bases and orthogonal families. The Gram-Schmidt method is presented as a technique to orthogonalize bases, ensuring linear independence. The process involves defining projections and iteratively constructing orthogonal vectors. The lecture also covers properties of orthogonal matrices and the importance of orthonormal bases. Various examples and applications are discussed, illustrating the significance of orthogonal bases in linear algebra.