\begin{questions}

\question{Show that a set of orthonormal vectors is linearly
independent.
}
\begin{solution}
Consider the set of orthonormal vectors $S=\left\lbrace e_j\right\rbrace_{j=0}^{\infty}$. Since it is orthonormal we have
\begin{align*}
\left<e_j,e_k\right>=\delta_{jk},
\end{align*}
where $\delta_{jk}$ is the Kronecker delta. Assume that 
\begin{align*}
\sum_{j=1}\alpha_je_j=0.
\end{align*}
Taking the inner product
\begin{align*}
0=\left<\sum_{j=1}\alpha_je_j,e_k\right>&=\sum_{j=1}\left<\alpha_je_j,e_k\right>\\
&=\sum_{j=1}\alpha_j\left<e_j,e_k\right>\\
&=\sum_{j=1}\alpha_j\delta_{jk}\\
&=\alpha_k.
\end{align*}
Hence, we have obtained that all the $\alpha_k=0$ and therefore it is proved that a set of orthonormal vectors is linearly independent.
\end{solution}
\end{questions}
