Referencia añadido: https://mathoverflow.net/questions/103152/determinant-of-integer-lattice-basis-of-l-x-1-ldots-x-n-a-1x-1-cdotsa
Sí, hay. Escribe tu objetivo de vectores como una fila $R$ fijo de números enteros. Requerimos su mcd 1. Ahora, mediante una secuencia de la columna de operaciones con matrices elementales $C_j,$ cada uno de todos los números enteros y cada determinante $1.$ El proceso resultante debe transformar $R$ a $(1,0,0).$ por otra parte, realmente habrá pocos $C_i,$ con el producto $C_1 C_2 C_3 ... = C.$, Entonces tenemos
$$ RC = (1,0,0). $$
Given this notation, the two desired vectors are the second and third column of $C.$ If we call those columns $v,w,$ then either $v \times w = R^T$ or $w \times v = R^T$
Let me make some examples...
$$
R =
\left(
\begin{array}{ccc}
3 & 5 & 7
\end{array}
\right)
$$
$$
C =
\left(
\begin{array}{ccc}
-2& 10& -7 \\
0 & 1 & 0 \\
1&-5&3 \\
\end{array}
\right)
$$
Como filas, tenemos la solución vectores
$$
\left(
\begin{array}{ccc}
10& 1& -5 \\
-7 & 0 & 3 \\
\end{array}
\right)
$$
Finding $C$
parisize = 4000000, primelimit = 500000
? r = [ 3,5,7]
%1 = [3, 5, 7]
? c1 = [ 1,0,-2;0,1,0; 0,0,1]
%2 =
[1 0 -2]
[0 1 0]
[0 0 1]
? r * c1
%3 = [3, 5, 1]
? c2 = [ 1,0,0;0,1,0; -3,-5,1]
%6 =
[ 1 0 0]
[ 0 1 0]
[-3 -5 1]
? r * c1 * c2
%7 = [0, 0, 1]
? c3 = [ 0,0,-1;0,1,0; 1,0,0]
%11 =
[0 0 -1]
[0 1 0]
[1 0 0]
? matdet(c3)
%12 = 1
? r * c1 * c2 * c3
%13 = [1, 0, 0]
? c = c1 * c2 * c3
%14 =
[-2 10 -7]
[ 0 1 0]
[ 1 -5 3]
? r * c
%15 = [1, 0, 0]
? matdet(c)
%16 = 1
?
P.S. If $g = \gcd(a,b,c) > 1,$ solve the problem for $\left(\frac{a}{g}, \frac{b}{g}, \frac{c}{g} \right).$ After finding $v,w,$ multiply one of them by $g$ but not the other.
============================================================================
$$
R =
\left(
\begin{array}{ccc}
6 & 10 & 15
\end{array}
\right)
$$
$$
C =
\left(
\begin{array}{ccc}
1& -10& -15 \\
1 & -9 & -15 \\
-1&10&16 \\
\end{array}
\right)
$$
Como filas, tenemos la solución vectores
$$
\left(
\begin{array}{ccc}
-10& -9& 10 \\
-15 & -15 & 16 \\
\end{array}
\right)
$$
parisize = 4000000, primelimit = 500000
? r = [ 6,10,15]
%1 = [6, 10, 15]
? c1 = [ 1,0,0; 1,1,0; -1,0,1]
%2 =
[ 1 0 0]
[ 1 1 0]
[-1 0 1]
? matdet(c1)
%3 = 1
? r * c1
%4 = [1, 10, 15]
? c2= [ 1,-10,-15; 0,1,0; 0,0,1]
%5 =
[1 -10 -15]
[0 1 0]
[0 0 1]
? r * c1 * c2
%6 = [1, 0, 0]
? c = c1 * c2
%7 =
[ 1 -10 -15]
[ 1 -9 -15]
[-1 10 16]
? matdet(c)
%8 = 1
?
===============================================================
Added: we could also write the thing out in symbols, given $\gcd(a,b,c) = 1,$ with $g = \gcd(b,c) $ and Bezout expressions $sb+tc= g$ and $pa+qg=1.$ A continuación, el método que nos conduce a la solución general
$$
\left(
\begin{array}{ccc}
-g& sa& ta \\
0 & -\frac{c}{g} & \frac{b}{g} \\
\end{array}
\right)
$$
Con el ejemplo
$$
R =
\left(
\begin{array}{ccc}
3 & 5 & 7
\end{array}
\right)
$$
llegamos $a=3, b=5, c=7, g=1, s=3,t=-2,q=1 $ y
$$
\left(
\begin{array}{ccc}
-1& 9& -6 \\
0 & -7 & 5 \\
\end{array}
\right)
$$
En esta dimensión podemos usar Gauss reducción de encontrar a una reducción de la base,
$$
\left(
\begin{array}{cc}
1& 1 \\
3 & 4 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
-1& 9& -6 \\
0 & -7 & 5 \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
-1& 2& -1 \\
-3 & -1 & 2 \\
\end{array}
\right)
$$
