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I have $k$ linearly independent vectors in $\mathbb{R}^n$. I want to know if the span of these vectors (i.e. the set of points in $\mathbb{R}^n$ that can be described by linear combinations of these vectors) intersects the portion of $\mathbb{R}^n$ where all the axes are positive (e.g. the first quadrant in $\mathbb{R}^2$, the first octant in $\mathbb{R}^3$, etc.).

Is there a test I can run on my vectors that will answer this question?

flag	asked Feb 29 2012 at 22:08 GB 368●1●6

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Just a sidenote: the general term for quadrant, octant, etc. which applies for any $n$ is "orthant". – Noah Stein Mar 1 2012 at 18:25

Igor Rivin · Answer 1 · 2012-02-29 22:34:46Z UTC

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The magic words are "Farkas' Lemma".

answered Feb 29 2012 at 22:34

Igor Rivin
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The specific form of the question (where the cone is exactly the positive orthant) shows up in option-pricing models. In Ross's textbook, An Elementary Introduction to Mathematical Finance, the exact question is called the Arbitrage Theorem. – Anthony Quas Mar 1 2012 at 0:10

Robert Israel · Answer 2 · 2012-03-01 17:15:25Z UTC

In practice, the way to do this (where $n$ and $k$ may be large) is with linear programming software, since this is basically a linear programming feasibility problem: $A x \ge (1,\ldots, 1)^T$ where $A$ is the $n \times k$ matrix whose columns are your vectors.

How do you tell if the span of a set of vectors enters the most positive sector of a graph?

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