4
$\begingroup$

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, $\vec{y}=\vec{x}\backslash x_k$?

$\endgroup$
  • 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
6
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.