# Linear Algebra/Orthogonality

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## Cauchy-Schwarz inequality

[edit | edit source]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

[edit | edit source]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 | edit source]## Applications

[edit | edit source]### Linear least squares

[edit | edit source]### How to orthogonalize a basis

[edit | edit source]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-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.