Is there an algorithm to get a set of $n$ distinct vectors which generates the given $n$ vectors, whose degree follows the irreducible factorization?

Question

For example, for $A=\begin{pmatrix}2&1\\0&2\end{pmatrix}$, its smith normal form is $\begin{pmatrix}1&0\\0&4\end{pmatrix}$. So I can calculate the smith normal form, or the irreducible factorization of given $n$ vectors. However, I couldn't find which vectors have order of such factorization, 4 in this example. Of course, this example is easy enough and I can easily give a vector $\begin{pmatrix}1&1\end{pmatrix}$ which 4 is the smallest natural number which $n\begin{pmatrix}1&1\end{pmatrix}\in \{v\in\mathbb{R}^2|v=k_1\begin{pmatrix}2&1\end{pmatrix}+k_2\begin{pmatrix}0&2\end{pmatrix},k_1,k_2\in\mathbb{Z}\}$. How do I find such vectors in general, assuming all $n$ vectors are in $\mathbb{R}^n$ and are linearly independent?