For an n-dimensional vector $v$, we define a truncated shift as \begin{equation} y = S_T v, \end{equation} where $S_T=[\delta_{i-j+T}]_{i, j=1}^n$.
Can we give an example of $v$ in which every $n$ non-zero distinct shifts of $v$ are linearly independent? (i.e., generates bases of $\mathbb{R}^n$)
Obviously vectors like $v=[1, 0, \ldots, 0]^T$ or $v=[1, 2, 0, \ldots, 0]^T$ are such trivial examples. However, I want to know about other (preferably structured) vectors with fewer zeros in them.
This example helps us to prove there exists an $m\times n$, $(n-m+1)$-banded Toeplitz matrix with SPARK=$n-m+1$. Further, we can utilize this fact to prove that if the elements of this banded Toeplitz are randomly generated with an absolutely continuous distribution, then, with probability 1 the Toeplitz matrix is full-spark (i.e., SPARK=$n-m+1$).