# Definition:Orthogonal (Linear Algebra)

## Definition

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

Let $u, v \in V$.

We say that $u$ and $v$ are orthogonal if and only if:

$\innerprod u v = 0$

We denote this:

$u \perp v$

### 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:

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

### Orthogonality of Sets

Let $A, B \subseteq V$.

We say that $A$ and $B$ are orthogonal if and only if:

$\forall a \in A, b \in B: a \perp b$

That is, if $a$ and $b$ are orthogonal elements of $A$ and $B$ for all $a \in A$ and $b \in B$.

We write:

$A \perp B$

### Orthogonal Complement

Let $S\subseteq V$ be a subset.

We define the orthogonal complement of $S$ (with respect to $\innerprod \cdot \cdot$), written $S^\perp$ as the set of all $v \in V$ which are orthogonal to all $s \in S$.

That is:

$S^\perp = \set {v \in V : \innerprod v s = 0 \text { for all } s \in S}$

If $S = \set v$ is a singleton, we may write $S^\perp$ as $v^\perp$.

### 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.