Orthogonality

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Definition

Two vectors $x$ and $y$ in an inner product space are orthogonal if their inner product is zero:

$$ \langle x, y \rangle = 0 $$

We write $x \perp y$ to denote orthogonality.

For Functions

Two functions $f$ and $g$ are orthogonal if:

$$ \langle f, g \rangle = \int f(x) \overline{g(x)} \, dx = 0 $$

Intuition

Orthogonal vectors are “perpendicular” — they point in completely independent directions. In the function space context, orthogonal functions have no “overlap” or “similarity.”

Examples

  • In $\mathbb{R}^3$: $(1, 0, 0) \perp (0, 1, 0)$
  • $\sin(nx)$ and $\sin(mx)$ are orthogonal on $[0, 2\pi]$ when $n \neq m$
  • $\sin(nx)$ and $\cos(mx)$ are orthogonal on $[0, 2\pi]$ for all $n, m$

Orthonormal Basis

A set of vectors $\{e_1, e_2, \ldots\}$ is orthonormal if:

  • $\langle e_i, e_j \rangle = 0$ for $i \neq j$ (orthogonal)
  • $\langle e_i, e_i \rangle = 1$ for all $i$ (normalized)

Fourier series use orthonormal bases of sines and cosines.