# Definition:Orthogonal (Linear Algebra)

This page needs proofreading.review the links here, and on subpagesIf you believe all issues are dealt with, please remove `{{Proofread}}` from the code.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Proofread}}` from the code. |

## 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.

## Also see

## Sources

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.will have ballsed up the flow for Conway in refactoring, sorryIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next): $\text I.2.1$ - 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions - 2020: James C. Robinson:
*Introduction to Functional Analysis*... (previous) ... (next) $9.2$: Orthonormal Sets