Linear Algebra/Inner Product Length and Orthogonality

Orthogonality

Cauchy-Schwartz inequality

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: $| \langle x,y \rangle | \le \| x \| \| y \|$.

Definition

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

Applications

How to orthogonalize a basis

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 { v₁, ... }.
Solution: Gram-Schidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.