Let $V\subseteq\mathbb Q^m$ be a vector space. A full rank set of indices for $V$ is a subset of $\{1,\dots,m\}$ of cardinal $d$ such that there exists a basis $\{v_1,\dots,v_d\}$ of $V$ with $v_i[i]=1$ and $v_i[j]=0$ for $i\ne j\in I$. The set $\{v_1,\dots,v_d\}$ is called a vector $I$-representation of $V$.
I would like to know whether the notions of «full rank set of indices for $V$» and of «vector $I$-representation of $V$» are already known (may be under another name) in some published paper/book. Or whether they are standard (I can't find reference on google, but those words are so common that it does not mean anything).
To be more precise, I found those names and definition in an unpublished paper, with some interesting propositions such as: each vector space admits a vector $I$-representation for some $I$ and its binary representation is at most polynomially bigger than the one of any other base, and it is polynomial-time computable from any basis of $V$. The paper is «Least Significant Digit First Presburger Automata», 2006.