Linear Algebra/Orthogonal Sets

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Orthogonal Sets[edit]

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[edit]



\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.