Given constants $n$ and $\ell$, where $\ell < n$, suppose we have the following set of equations in column vectors $a_1, a_2, \dots, a_n \in {\mathbb{C}^{\ell}}$

$$\begin{array}{rl} \langle a_1, a_n \rangle &= 0\\ \langle a_1, a_{n-1} \rangle + \langle a_2, a_n \rangle &= 0\\ \langle a_1, a_{n-2} \rangle + \langle a_2, a_{n-1} \rangle + \langle a_3, a_{n}\rangle &= 0\\ &\vdots\\ \langle a_1, a_2 \rangle + \langle a_2, a_3 \rangle + \cdots + \langle a_{n-1}, a_n \rangle &= 0\end{array}$$

where $\langle a, b \rangle := a^H b$. How can one find $a_1, a_2, \dots, a_n$ that satisfy these equations?

don'tsee why it is trivial off hand. How about you, @GerryMyerson? (or any other one who voted to close). . $\endgroup$ – fedja Dec 5 '17 at 15:12