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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.

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  • $\begingroup$ Could you provide some context as to where you encountered this notion? Maybe a quote from a text? $\endgroup$ Jul 30, 2016 at 11:11
  • $\begingroup$ 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 $\endgroup$
    – Dilawar
    Aug 13, 2017 at 6:34

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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 $x+y=1$.
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