# Linear Algebra/Inner Product Length and Orthogonality

## Orthogonality

edit### Cauchy-Schwarz inequality

editThe 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

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

### Orthogonal complement and matrix transpose

edit### Applications

edit#### Linear least squares

edit#### How to orthogonalize a basis

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