Linear Algebra/Orthogonal Sets

Orthogonal Sets

Given a set $A = ( a_{1},a_{2}, \ldots ,a_{n} )$, 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

$\boldsymbol{\Omega} = ( \omega_{1}, \omega_{2}, \omega_{3} )$

$\omega_{1} = \begin{bmatrix} 1 \\ 0 \\ 2 \\ 1 \\ \end{bmatrix} , \omega_{2} = \begin{bmatrix} 2 \\ 3 \\ -2 \\ 2 \\ \end{bmatrix}, \omega_{3} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ -1 \\ \end{bmatrix}$

We see that

$\omega_1 \cdot \omega_2 = 0$

$\omega_1 \cdot \omega_3 = 0$

$\omega_2 \cdot \omega_3 = 0$

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