Definition:Orthogonal (Linear Algebra)

Definition
Let $\left({V, \left\langle {\cdot} \right\rangle}\right)$ 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 its elements are pairwise orthogonal, that is:


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

Vectors in $\R^n$
Let $\mathbf{u}$, $\mathbf{v}$ be vectors in $\R^n$.


 * $\mathbf{u}$ and $\mathbf{v}$ are said to be orthogonal iff their dot product is zero:


 * $\mathbf{u} \cdot \mathbf{v} = 0$

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

If in addition $\left\Vert {u_i}\right\Vert = 1$ for $i = 1, \ldots, n$, we call $S$ an orthonormal set.

Also see

 * Perpendicular
 * Non-Zero Vectors Orthogonal iff Pependicular