Example CROB4 Coordinatization relative to an orthonormal basis, $\complex{4}$

The set \begin{equation*} \set{\vect{x}_1,\,\vect{x}_2,\,\vect{x}_3,\,\vect{x}_4}= \set{ \colvector{1+i\\1\\1-i\\i},\, \colvector{1+5i\\6+5i\\-7-i\\1-6i},\, \colvector{-7+34i\\-8-23i\\-10+22i\\30+13i},\, \colvector{-2-4i\\6+i\\4+3i\\6-i} } \end{equation*} was proposed, and partially verified, as an orthogonal set in Example AOS. Let us scale each vector to norm 1, so as to form an orthonormal set in $\complex{4}$. Then by Theorem OSLI the set will be linearly independent, and by Theorem NME5 the set will be a basis for $\complex{4}$. So, once scaled to norm 1, the adjusted set will be an orthonormal basis of $\complex{4}$. The norms are, \begin{align*} \norm{\vect{x}_1}=\sqrt{6} && \norm{\vect{x}_2}=\sqrt{174} && \norm{\vect{x}_3}=\sqrt{3451} && \norm{\vect{x}_4}=\sqrt{119} \end{align*}

So an orthonormal basis is \begin{align*} B&= \set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3,\,\vect{v}_4}\\ &=\set{ \frac{1}{\sqrt{6}}\colvector{1+i\\1\\1-i\\i},\, \frac{1}{\sqrt{174}}\colvector{1+5i\\6+5i\\-7-i\\1-6i},\, \frac{1}{\sqrt{3451}}\colvector{-7+34i\\-8-23i\\-10+22i\\30+13i},\, \frac{1}{\sqrt{119}}\colvector{-2-4i\\6+i\\4+3i\\6-i} } \end{align*}

Now, to illustrate Theorem COB, choose any vector from $\complex{4}$, say $\vect{w}=\colvector{2\\-3\\1\\4}$, and compute \begin{align*} \innerproduct{\vect{v}_1}{\vect{w}}&=\frac{-5i}{\sqrt{6}}& \innerproduct{\vect{v}_2}{\vect{w}}&=\frac{-19+30i}{\sqrt{174}}\\ \innerproduct{\vect{v}_3}{\vect{w}}&=\frac{120-211i}{\sqrt{3451}}& \innerproduct{\vect{v}_4}{\vect{w}}&=\frac{6+12i}{\sqrt{119}} \end{align*}

Then Theorem COB guarantees that \begin{align*} \colvector{2\\-3\\1\\4}&= \frac{-5i}{\sqrt{6}}\left(\frac{1}{\sqrt{6}}\colvector{1+i\\1\\1-i\\i}\right)+ \frac{-19+30i}{\sqrt{174}}\left(\frac{1}{\sqrt{174}}\colvector{1+5i\\6+5i\\-7-i\\1-6i}\right)\\ &\quad\quad+ \frac{120-211i}{\sqrt{3451}}\left(\frac{1}{\sqrt{3451}}\colvector{-7+34i\\-8-23i\\-10+22i\\30+13i}\right)+ \frac{6+12i}{\sqrt{119}}\left(\frac{1}{\sqrt{119}}\colvector{-2-4i\\6+i\\4+3i\\6-i}\right) \end{align*} as you might want to check (if you have unlimited patience).