Every subgroup $H$ of a free abelian group $G$ of rank $n$ is free of rank $s \leq n$. Moreover there exists a basis $u_1, ... , u_n$ for $G$ and positive integers $\alpha_1, ... ,\alpha_s$ such that $\alpha_1 u_1, ... , \alpha_s u_s$ is a basis for $H$.
This is Theorem 1.16. in the book Algebraic-Number Theory by Ian Stewart and David Tall, on page 29. The first part of the proof is -
In the proof it is written that -
From all such coefficients, let, $\lambda(w_1 , ... , w_n)$ be the least positive integer occurring.
and
$u_1, w_2, ... , w_n$ is another basis for $G$. (The appropriate matrix is clearly unimodular.)
QUESTION
What is the meaning of "$\lambda(w_1 , ... , w_n)$ be the least positive integer occurring"? i.e. what is $\lambda(w_1 , ... , w_n)$?
How to show that the appropriate matrix of $u_1, w_2, ... , w_n$ is unimodular? Plz show the derivation, thanks.
Regarding the first question: $\lambda(w_1,\dots,w_n)$ is equal to the minimum element of the set of positive coefficients, i.e. $\min\{h_i : 1 \leq i \leq n, h_i > 0\}$.
Regarding the second question: the associated matrix is $$ \pmatrix{1&0&\cdots && 0\\ q_2 &1&\ddots&&\vdots\\ q_3 & 0 & \ddots&0&0\\ \vdots & \vdots& \ddots & \ddots & 0\\ q_n & 0 & \cdots &0& 1}. $$ There are many ways to see that this matrix is unimodular. One is to note first that the matrix is lower triangular.