# How can I indicate formally that an element is removed from a vector?

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

• 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. – Aidan Mar 26 at 16:42
• Thanks, @Aidan! – a25bedc5-3d09-41b8-82fb-ea6c353d75ae Mar 26 at 17:03
• I think it would be more fitting to talk about lists here, not “vectors”. – 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.