I understand that when a $m\times n$ matrix $A$ with $m>n$ has full rank, then the diagonal entries of its $R$ matrix from its reduced QR factorisation are all non-zero.
But I saw a literature online, which extended this; and said when only $'k'$ diagonal entries of matrix $R$ are non zero, then $k\leq\operatorname{rank}(A)< n $ ; I don't understand the proof of this statement.
From my understanding; each non-zero diagonal entry of $R$ from reduced QR factorisation of $A$, means that there is a vector in $\operatorname{col}(A)$ which is linearly dependent on the rest of the independent basis! Thus; we must have $n-k$ linearly dependent vectors in the $\operatorname{col}(A)$.