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 $i\neq j$.

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.