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: $|⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert$.

Definition

For any vectors $x$ and $y$ in an inner product space $V$, we say $x$ is orthogonal to $y$, and denote it by $x\mathrm{\perp }y$, if $⟨x,y⟩=0$.

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