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?