Definition:Orthogonal (Linear Algebra)

Definition
Let $\left({ V, \langle \cdot \rangle }\right)$ be an inner product space, and let $x, y \in V$.

We say that $u$ and $v$ are orthogonal if $\left\langle{ u, v }\right\rangle = 0$.

More generally, if $S = \left\{ u_1, \ldots, u_n \right\}$ is a subset of $V$, we that $S$ is an orthogonal set if it's elements are pairwise orthogonal, that is,


 * $ \left\langle u_i, u_j \right\rangle = 0,\quad \forall i \neq j$

Let $\| u \| = \sqrt{\left\langle u, u \right\rangle}$, $u \in V$ define the norm associated to $\left\langle \cdot \right\rangle$.

If in addition $\| u_i \| = 1$ for $i = 1, \ldots, n$, we call $S$ an orthonormal set.