Linear Algebra/Inner Product Length and Orthogonality

Orthogonality Cauchy-Schwarz inequality The Cauchy-Schwarz inequality states that the magnitude of the inner product of two vectors is less than or equal to the product of the vector norms, or: . Definition For any vectors and in an inner product space , we say is orthogonal to , and denote it by , if . Orthogonal complement and matrix transpose Applications Linear least squares How to orthogonalize a basis Suppose to be on a vector space V with a scalar product (not necessarily positive-definite), Problem: Construct an orthonormal basis of V starting by a random basis { v1, ... }. Solution: Gram-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate. Linear Algebra