I have a question related to geometry of numbers, which although seems quite basic, I was rather confused by it so I decided to ask here. Let $\Lambda$ be a lattice in $\mathbb{R}^n$. Let $R_1, ..., R_n$ be the successive minima of $\Lambda$ and $\mathbf{x}^{(1)}, ..., \mathbf{x}^{(n)}$ be the minimal points of $\Lambda$.

In "Analytic Methods for Diophantine Equations and Diophantine Inequalities", by Harold Davenport, in Lemma 12.2, he proves that if $d(\Lambda)=1$, then $$ 1 \leq R_1...R_n \leq 2^nJ_n, $$ where $J_n$ is the volume of a sphere of radius $1$ in $n$ dimensions. The very first step of the proof is to "rotate the $n$-dimensional space about $O$ until" the matrix $[\mathbf{x}^{(1)}, ..., \mathbf{x}^{(n)}]$ is upper triangular.

What I was wondering was that in stead of rotation, could we instead: change coordinates via elementary row operations so that, in the new coordinate system, the minimal points have the given shape? (Since elementary row operations have a non-zero determinant, we can divide the resulting transformation by their determinants to obtain a transformation which has determinant $1$.)

I felt perhaps there may be some issues with this, but I was not quite sure. I would appreciate any clarification. Thank you!