Linear Algebra/Orthogonal Sets

Orthogonal Sets

Given a set $A=(a_1,a_2,\dots ,a_n)$, where $a_1$ through $a_n$ are nonzero vectors of the same dimension, is an orthogonal set if

$a_i\cdot a_j=0$

where $i\ne j$.

So, for example, if one has a set of 3 vectors with the same dimension (for example $4\times 1$) 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

$\mathrm{\Omega }=({\omega }_1,{\omega }_2,{\omega }_3)$

${\omega }_1=\left[\begin{array}{c}1\\ 0\\ 2\\ 1\end{array}\right],{\omega }_2=\left[\begin{array}{c}2\\ 3\\ -2\\ 2\end{array}\right],{\omega }_3=\left[\begin{array}{c}1\\ 0\\ 0\\ -1\end{array}\right]$

We see that

${\omega }_1\cdot {\omega }_2=0$

${\omega }_1\cdot {\omega }_3=0$

${\omega }_2\cdot {\omega }_3=0$

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