Suppose I have an input set of integer vectors, $S = \{v_1,v_2,...,v_n\}$. How do I test if another integer vector $w$ is in the integer span of $S$?
Test if a vector is in integer span of vectors
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linear-algebra
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0You solve the linear system and check whether it has integer solutions. – 2017-02-10
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0Sure, but how do I check if it has integer solutions? – 2017-02-10
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0Well, when you have found the solutions… – 2017-02-10
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0By inspection. If the coordinates of everyone there are integers and the determinant of the coefficient matrix is $1$ or $-1$, then the inverse matrix will have integer entries. – 2017-02-10
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0It's not as trivial as the other comments would have you believe; the $1$-dimensional case amounts to the study of a linear diophantine equation. – 2017-02-10
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0@Student what you're considering is a system of linear diophantine equations. Wikipedia has a discussion [over here](https://en.wikipedia.org/wiki/Diophantine_equation#System_of_linear_Diophantine_equations). – 2017-02-10
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0My intuition is that it should be solving an integer linear program without a target function, but there ought to be an easier way. – 2017-02-10