A pair of vectors in an inner product space are orthogonal if their inner product is zero. This could be useful, for example, to check for linear independence.
Definition
Orthogonal Vectors & Sets ^formula1
A pair of vectors, in an inner product space are orthogonal iff:
\langle \vec{u}, \vec{v} \rangle = 0 \quad\quad \vec{u} \neq \vec{v}$$ Note that in a *complex* inner product space, two vectors can be orthogonal and *not* be at 90 degree angles, since the [definition for angles only checks real components](Geometry%20from%20Inner%20Products.md#Angle) \ A [set](Set%20-%20Maths.md) of vectors, $\{ \vec{v}_{1}, \vec{v}_{2} , \dots \vec{v}_{n} \}$ is **orthogonal** iff:\langle \vec{v}{1} \vec{v}{j}\rangle = 0 \quad\quad i \neq j, i,j \in [0, n]
We also have a subcategory of orthogonal vectors, known as orthonormal vectors.
Orthonormal Vectors & Sets
Orthonormal Vectors & Sets ^formula2
If all of these vectors have a norm of 1, then the set is orthonormal:
\langle \vec{v}_{i}, \vec{v}_{j} \rangle = \begin{cases}
0, & i \neq j \ 1, & i = j \end{cases}
>\
>Similarly, a set of vectors is **orthonormal** if every pair of vectors are orthonormal.
### Properties
>[!theorem] Linear Independence of Orthogonal Sets ^theorem
>
>Every **orthogonal set** of non-zero vectors in an [inner product space](Inner%20Product.md) is [linearly independent](Linear%20Dependence.md).
### Applications
##### Expressing a vector in a vector space
If the [basis](Basis.md) of an [inner product space](Inner%20Product.md) is an [orthogonal](Orthogonality.md) [set](Set%20-%20Maths.md), then we can easily represent any vector in the space using the [inner product](Inner%20Product.md):
>[!theorem] Vector expressed through an orthonormal basis ^theorem
>
>If an [inner product space](Inner%20Product.md), $V^{n}$, has an *ordered*, [orthogonal](Orthogonality.md) [basis](Basis.md), $\mathcal{B} = \{ \vec{b}_{1}, \vec{b}_{2}, \dots \vec{b}_{n} \}$, then:
>$$
>\vec{v} = \langle \vec{v}, \vec{b}_{1} \rangle \vec{b}_{1} + \langle \vec{v}, \vec{b}_{2} \rangle \vec{b}_{2} + \dots + \langle \vec{v}, \vec{b}_{n} \rangle \vec{b}_{n}
>$$
>I.e. a vector can be expressed using the inner product.
##### Pythagorean Theorem
Using orthogonal vectors, we can extend the [Pythagorean theorem](Pythagorean%20Theorem.md) to work for all real [inner product spaces](Inner%20Product.md):
#TODO