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.