# General Position

What does it mean when the columns of a matrix are in general position? I do not know if this is relevant or not, but the matrix in question is under-determined.

• Could you provide some context as to where you encountered this notion? Maybe a quote from a text? Jul 30, 2016 at 11:11
• I guess OP is referring to subject of 'Compressed Sensing'. See this paper D.L. Donoho, “Compressed sensing,” IEEE. Trans. Info. Thry., vol. 52, no. 4, pp. 1289–1306, 2006 Aug 13, 2017 at 6:34

I have never heard it used quite this way, but here's what is most probably meant. Let's say we have an $(m\times n)$-matrix $A$. Then the columns of $A$ can be considered as points in $\mathbb{R}^m$. Let's call these points $p_1, \ldots, p_n$. Then by the columns of $A$ are in general position, what is probably meant is that the points $p_1, \ldots, p_n$ are in general linear position.

The points $p_1, \ldots, p_n$ are in general linear position if they are not all in the same $(n-2)$-dimensional affine subspace.

Examples:

• the points $(1,0)$ and $(0,1)$ are in general position, because they are not equal;
• the points $(1,0)$, $(\frac12,\frac12)$ and $(0,1)$ are not in general position, because they are contained in the line $x+y=1$.