# Linear Algebra/Orthogonal Sets

## Orthogonal Sets

Given a set ${\displaystyle A=(a_{1},a_{2},\ldots ,a_{n})}$, where ${\displaystyle a_{1}}$ through ${\displaystyle a_{n}}$ are nonzero vectors of the same dimension, is an orthogonal set if

${\displaystyle a_{i}\cdot a_{j}=0}$

where ${\displaystyle i\neq j}$.

So, for example, if one has a set of 3 vectors with the same dimension (for example ${\displaystyle 4\times 1}$) and taking the dot product of each vector with each other vector all equal zero, it is an orthogonal set. This is illustrated below.

##### Example of Orthogonal Set

${\displaystyle {\boldsymbol {\Omega }}=(\omega _{1},\omega _{2},\omega _{3})}$

${\displaystyle \omega _{1}={\begin{bmatrix}1\\0\\2\\1\\\end{bmatrix}},\omega _{2}={\begin{bmatrix}2\\3\\-2\\2\\\end{bmatrix}},\omega _{3}={\begin{bmatrix}1\\0\\0\\-1\\\end{bmatrix}}}$

We see that

${\displaystyle \omega _{1}\cdot \omega _{2}=0}$

${\displaystyle \omega _{1}\cdot \omega _{3}=0}$

${\displaystyle \omega _{2}\cdot \omega _{3}=0}$

Thus, ${\displaystyle {\boldsymbol {\Omega }}}$ is an orthogonal set of vectors.