# Definition:Orthogonal (Linear Algebra)

## Definition

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $u, v \in V$.

Then $u$ and $v$ are orthogonal if and only if:

$\innerprod u v = 0$

### Orthogonal Set

Let $S = \set {u_1, \ldots, u_n}$ be a subset of $V$.

Then $S$ is an orthogonal set if and only if its elements are pairwise orthogonal:

$\forall i \ne j: \innerprod {u_i}, {u_j} = 0$

### Vectors in $\R^n$

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

Then $\mathbf u$ and $\mathbf v$ are said to be orthogonal if and only if their dot product is zero:

$\mathbf u \cdot \mathbf v = 0$

As Dot Product is Inner Product, this is a special case of the definition of orthogonal vectors.

## Sources

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