How should the sequence or list $k+1, k+2, \ldots, m$ be interpreted in a proof when $k \ge m$ ?
Context:
Suppose the matrix $K$ ($m \times i$) has $k$ pivots and let $q$ be the first column of the matrix $L$ ($m \times n-i$).
If $q_{k+1}, q_{k+2} \ldots, q_{m}$ are all zero, then $K^{'}=[K|q]$ is in RREF.
If $q_l \neq 0$ with $k+1 \le l \le m$ then a matrix $B$ is made $...$.
What if $k = m$ then none of the cases happen ?
The list or sequence $(x_{k+1},...,x_m),$ when $k\geqslant m,$ is the empty list or sequence. This is a well defined mathematical object, as useful as, and analogous to, zero or the empty set.
In general, a sequence, list, or string can be thought of as an initial segment of a map from the positive integers to a set in which the terms reside. Such a map is a set of ordered pairs; so the empty sequence is just the empty set $\varnothing$. Zero is also the empty set in the standard (Von Neumann) construction of the natural numbers. However, it is conventional not to use $\varnothing$ or $0$ to denote the empty sequence, because operations on sequences (e.g. concatenation and truncation) differ from those done on sets (e.g. union and set-subtraction) or numbers (e.g. addition and subtraction)---and, in particular, we often want to use $0$ as a potential term of a sequence. If you need a symbol to denote the empty sequence, I suggest $\epsilon$ (I prefer $\varepsilon$ for "small" quantities), if this is not needed for other things; $\Lambda$ is also used.
Mathematicians often don't bother with set-theoretically pinned-down definitions of the elementary objects in their field: "as long as you have a clear intuition of it, the set-theoretic nuts and bolts are just a distraction" is the general idea. So you might not easily find the definition in the literature.
Referring to the clause in question in your post, in the case $k\geq m\,$: "If $q_{k+1},q_{k+2},…,q_m$ are all zero, ..." should be interpreted as "If every term of the (empty) sequence $(q_{k+1},q_{k+2},…,q_m)$ is zero, ...". This is known as a vacuous condition: it is automatically true. (Proof: give me a term of the sequence, and I can show you that it equals $0$.) So effectively you can delete that condition, and the rest of the statement becomes unconditional. Whether that was the author's intention is another matter.
If $k\ge m$, then the set $\{k+1,k+2,\dots,m\}=\{i\,:\,k<i\le m\}$ is empty.
Therefore I suppose there is a typo in your statement.
