Orthogonality

A rigorous guide to inner products, orthogonal sets, complements, and orthonormal coordinates.

Orthogonal projection separates a vector into fitted and perpendicular components. A vector is decomposed into a component along a subspace and a right-angled residual. Basis arrows show how coordinates reconstruct the projected result.

Precise definition

Vectors $u,v$ are orthogonal when $\langle u,v\rangle=0$. A set is orthogonal when distinct vectors are pairwise orthogonal, and orthonormal when additionally every vector has norm 1. Any orthogonal set of nonzero vectors is linearly independent.

Notation and mathematical language

For a subspace $W$, $W^\perp=\{v:\langle v,w\rangle=0\text{ for all }w\in W\}$. In $\mathbb R^n$, $\operatorname{Row}(A)^\perp=\ker A$ and $\operatorname{Col}(A)^\perp=\ker A^T$.

Conceptual picture

Orthogonal directions do not interfere in the inner product. Pythagoras gives $\|u+v\|^2=\|u\|^2+\|v\|^2$ when $u\perp v$. Orthonormal coordinates satisfy $c_i=\langle v,q_i\rangle$.

Fully worked example

Interpretation and application

Orthogonality simplifies energy, Fourier analysis, regression, error decomposition, and numerical algorithms. Unit choices affect a raw dot product, so applications may need nondimensionalization or a physically meaningful weighted inner product.