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Given a vector $v=<1,2,3>$ I want to have a new vector $v'$, which is the vector $v$, appends with a number $4$. How should I represent $v'$ mathematically?

What I wish to have is something like $v'=v^4=<1,2,3,4>$, where ^ is an appending symbol for vector (therefore my main question is, does it exist such "vector appending" symbol in maths?)

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What do you mean by appending? What is "type alphabet"? – EuYu Nov 15 '12 at 2:18
This question has also been asked at tex.SE: – Douglas S. Stones Nov 15 '12 at 3:02
We rarely append vectors. We do append other things (like words, although we use the fancier verb «to yuxtapose») and then we use no sign, just like multiplication. – Mariano Suárez-Alvarez Feb 1 '15 at 3:39

I usually see it being represented by writing $v=\left[1, 2, 3\right]$, $w=\left[4\right]$, and then $$\left[v,w\right]=\left[v,4\right]=\left[1,2,3,4\right]$$

This is similar to MATLAB notation, so it might be more common in computing-related contexts.

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This is a natural application of block matrix notation. If we can write $$\begin{bmatrix}\mathbf A&\mathbf b\\\mathbf b^T&c\end{bmatrix}$$ where $\mathbf A$, $\mathbf b$, and $c$ are a matrix, a column vector, and a scalar respectively, then surely we can write $$\begin{bmatrix}\mathbf v\\w\end{bmatrix}$$ to denote $$\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\\w\end{bmatrix}$$ (and we do). – Rahul May 19 '15 at 14:40

A fairly common notation for the concatenation of two sequences is $s^\frown t$. In the special case where $t$ consists of a single element $a$ you'd have $s^\frown\langle a\rangle$. The TeX code for the symbol $\frown$ is \frown, and to raise it to the desired level you'd make it a superscript.

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