I am looking for a rule or condition to create a set of linearly independent vectors that are not all orthogonal to each other.
Consider the following context: I have a team of $M \; (<n)$ members and each of them has to give me a vector $$v_i \in \mathbb{C}^n$$ in sequence, like the first member give a vector then second and so on. But I don't want the $k$th member to give a vector that is orthogonal to all the vectors given before him, but it should be linearly independent. So I want to impose a rule that every member should follow in order to create these vectors. We can think of these vectors as encoded messages.
One way I thought of is to fix one coordinate say first one to $1$ and for rest $n-1$ coordinate we ask them to create orthogonal vectors. But this way I am wasting one coordinate, I am looking for some more creative way to do this.
The chance of finding a random vector, that is linearly dependent w.r.t a family of vectors is 0. The chance of finding a vector, that is orthogonal w.r.t a family of vectors is also 0. Just letting each member guess a random vector should do the trick. If you want to be totally sure, check and let the unlucky person do a reroll.
This paper estimates the condition number of a matrix (somewhat a measure of orthogonality of the column vectors) to be logarithmic in the number of vectors ($1$ means orthogonal, $\infty$ means linearly dependent).
For a more algorithmic approach, each one could guess a random vector, orthogonalize w.r.t previous vectors and then adding older vectors back. This is essentially finding an orthogonal base and multiplying it with a triangular matrix. This is somewhat reversing the QR-decomposition.