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Given $E$ a finite subset of a real vector space $V$, a circuit of the associated matroid is a minimal linearly dependent subset of $E$. For each circuit $\underline X$, a minimal linear dependence $$\sum_{x\in \underline X}\lambda_xx=0$$ gives rise to a signed circuit $X = (X^+,X^-)$ defined by $$X^+= \{x\in \underline X: \lambda_x>0\}$$ and $$X^-=\{x\in \underline X:\lambda_x<0\}.$$ (So $(X^-,X^+)$ is also a signed circuit.)

Given an arbitrary linear dependence of the form $$\sum_{v\in S\subset E}\lambda_v v = 0$$ with $\lambda_v\neq 0$ for all $v\in S$, by definition there is a circuit $\underline X$ with $\underline X\subset S$.

My question is: if we define $S^+ = \{v\in S:\lambda_v>0\}$ and $S^-=\{v\in S:\lambda_v<0\}$, must there exist a signed circuit $X$ with $X^\pm\subset S^\pm$?

If it's useful or necessary, for my purposes I am only interested in situations where all coefficients are integers. Thanks in advance.

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I reasked this question on MathOverflow and Jan Kyncl gave a nice proof that there does indeed exist such a circuit:

https://mathoverflow.net/questions/157306/circuits-in-a-linear-oriented-matroid

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