Consider $\vec{x}\in\mathbb{R}^n$ and $\vec{y}\in\mathbb{R}^{n-1}$, where $ \vec{y} = (x_1, \dots, x_{k-1}, x_{k+1}, \dots, x_n)^\intercal $ $\forall\;x_k\in\vec{x}$.

How can I formally write that $\vec{y}$ is composed of all elements of $\vec{x}$, but does not contain the $k$-th one?

Is it possible to write it in terms of a set? For example, $\vec{y}=\vec{x}\backslash x_k$?

  • 4
    $\begingroup$ I've seen $(x_1,\ldots,\hat{x}_k,\ldots,x_n)$ as a fairly common convention, but whatever notation you use, you will want to define it. $\endgroup$ – Aidan Mar 26 at 16:42
  • $\begingroup$ Thanks, @Aidan! $\endgroup$ – a25bedc5-3d09-41b8-82fb-ea6c353d75ae Mar 26 at 17:03
  • $\begingroup$ I think it would be more fitting to talk about lists here, not “vectors”. $\endgroup$ – leftaroundabout Mar 27 at 0:55

There's no standard notation for this. I think the most common way is to do exactly what you did: write it out omitting an index $k$ — this way it's perfectly clear what you mean. The second most common option that I've seen in many publications is to use a hat to denote a dropped component, but still the author would always say explicitly what this hat means (precisely because it's not standard enough). So it would look something like this:

For a vector $\vec{x}\in\mathbb{R}^n$, consider the vector $\vec{y}\in\mathbb{R}^{n-1}$, $\vec{y}=(\dots,\widehat{x_k},\dots)^\intercal$, where the hat indicates omitting the $k$-th component.


Keep in mind that you, as an author, get to define your own notation. If you feel like $y = (x_1,x_2,\dots,x_{k-1},x_{k+1},\dots,x_n)$ is too lengthy, consider using something like Matlab notation $y=(x_{1:k-1},x_{k+1:n})$, or set notation $y=x_{\mathcal{I}\setminus \{k\}}$, where $\mathcal{I} = \{1,2,\dots,n\}$. Both of these allow you to clearly and quickly write down more complicated removals, such as the vector indexed by even indices: $z = x_{2:2:n}$ or $z = x_{\mathcal{I}\cap2\mathbb{N}}$. Whatever notation you choose to use, be sure to clearly explain it before using it.


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