Talk:Spark (mathematics)

The article says, "If all the columns are linearly independent, ${\displaystyle \mathrm {spark} (A)}$ is usually defined to be ${\displaystyle m+1}$". However, I'm questioning this claim. For instance, I wonder if it "should" instead be e.g. ${\displaystyle n+1}$. The two supporting references do not seem relevant; the Princeton Companion to Applied Mathematics article does not consider the full-column-rank case, and the Industrial Mathematics and Complex Systems chapter does not apply. Thatsme314 (talk) 22:18, 8 May 2024 (UTC)

I agree. There is an m+1 lower in the article that actually does make sense, and I imagine that this m+1 occurred as an accidental result of copying that other m+1. Other sources like this one say that it should be n+1, and that makes much more sense. 76.11.97.150 (talk) 17:56, 4 June 2024 (UTC)

I think it’s just a matter of perspective. For an mxn matrix with linearly independent columns, the column vectors are ostensibly members of two separate vector spaces: m dimensional Euclidean space and the n dimensional span of the columns. The point of the definition in these cases is merely to show that you need more vectors than you have columns in order to make them dependent. If you think about them as being in m dimensional space, then you should define it to be m+1. If you think of it as the span of the columns, it should be n+1. The same theorems hold in both cases, and the definitions match when you have the maximum amount of linearly independent columns. 14lclark (talk) 21:01, 28 December 2025 (UTC)