Orthogonality

In a vector space equipped with an inner product $\langle\cdot,\cdot\rangle$, two vectors $u$ and $v$ are orthogonal if

A set $\{v_i\}_{i\in I}$ is orthogonal if $\langle v_i,v_j\rangle=0$ for all $i\neq j$, and it is orthonormal if it is orthogonal and additionally $\|v_i\|=1$ for all $i$, where $\|v\|=\sqrt{\langle v,v\rangle}$ (compare the Euclidean norm in $\mathbb{R}^k$).

Orthogonality depends on the chosen inner product; changing the inner product can change which vectors are orthogonal.

Examples:

• In Euclidean space $\mathbb{R}^k$ with the standard inner product, the standard basis vectors $e_1,\dots,e_k$ form an orthonormal set.
• In $\mathbb{R}^2$, the vectors $(1,1)$ and $(1,-1)$ are orthogonal for the standard dot product.
• In $\mathbb{C}^2$ with the standard Hermitian inner product, $(1,0)$ is orthogonal to $(0,1)$.