# Linear Algebra/Inner Product Length and Orthogonality

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## Contents

## Orthogonality[edit]

### Cauchy-Schwarz inequality[edit]

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: .

### Definition[edit]

For 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[edit]

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.