View source for Linear Algebra/Inner Product Length and Orthogonality

== Orthogonality ==

=== Cauchy-Schwarz inequality ===

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:
<math>| \langle x,y \rangle |  \le  \| x \| \| y \| </math>.

=== Definition ===

For any vectors <math>x</math> and <math>y</math> in an inner product space <math>V</math>, we say <math>x</math> is orthogonal to <math>y</math>, and denote it by <math>x \bot y</math>, if <math>\langle x,y \rangle =0</math>.

=== Orthogonal complement and matrix transpose ===

=== Applications ===

==== Linear least squares ====

==== How to orthogonalize a basis ====
Suppose to be on a vector space V with a scalar product (not necessarily positive-definite), 
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'''Problem:''' Construct an orthonormal basis of ''V'' starting by a random basis ''{'' ''v''<sub>''1''</sub>, ... ''}''.
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Solution: Gram-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.
{{Subject|Linear Algebra}}

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