# 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 *{* *v*_{1}, ... *}*.

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