I'm trying to understand how the LLL algorithm works and I've stumbled upon the following question:
Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process obtaining $(v_{1},\cdots,v_{n})$.
Now I'd like to swap position $i$ and $i+1$ in $(b_1,\cdots,b_n)$ resulting into performing Gram Schmidt process on $B' = (b_1,\cdots,b_{i+1},b_{i},\cdots, b_n)$. According to this property of orthogonal transformation it should follow that $||v_{i}|| = ||v'_{i}||$ for each $i$, but the focus in my notes it that $||v_{i}||\cdot ||v_{i+1}|| = ||v'_{i}||\cdot||v'_{i+1}||$, which should trivially true since they are term by term equal.
What am I missing here? Any help would be appreciated.