Linear Algebra/Inner Product Length and Orthogonality

OrthogonalityEdit

Cauchy-Schwarz inequalityEdit

The Cauchy-Schwartz 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:  .

DefinitionEdit

For any vectors   and   in an inner product space  , we say   is orthogonal to  , and denote it by  , if  .

Orthogonal complement and matrix transposeEdit

ApplicationsEdit

Linear least squaresEdit

How to orthogonalize a basisEdit

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-Schidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.