Math notation to describe extending a vector? I have vectors $\vec{x} \in \mathbb{R}^4, \vec{y} \in \mathbb{R}^2, \vec{z} \in \mathbb{R}^2$, how can I describe that $x$ is simply the concatenation of $y$ and $z$? 
I want to convey that we take the elements of $\vec{y}$ and put them above the elements of $\vec{z}$ to get $\vec{x}$.
$\vec{x} = \vec{y} + \vec{z}$ is definitely wrong!
What's the terminology for this operation? extension or concatenation? Google really couldn't help!
 A: I don't know of a completely standard method of expressing this, but some things I have seen are $x = (y,z)$ (seeing $\mathbb{R}^4$ as the direct product of two copies of $\mathbb{R}^2$) and $x=\begin{pmatrix} y\\z\end{pmatrix}$ (thinking in terms of block matrices, assuming each vector is written as a column vector).
A nonstandard but suggestive notation could be $x = y\oplus z$, thinking of $\mathbb{R}^4$ as the direct sum of two copies of $\mathbb{R}^2$. I do not believe it is conventional to write elements of direct sums in this way, but it is conventional to do the perfectly analogous thing with tensor products, so I'd make the case that it's a good notation.
(To be specific: in the category of $\mathbb{R}$-algebras, tensor product is the coproduct. If $X,Y$ are two $\mathbb{R}$-algebras, their tensor product is written $X\otimes Y$. As with all coproducts, $X$ and $Y$ each have a canonical map into $X\otimes Y$ and if $x\in X, y\in Y$, the product of the canonical images of $x,y$ in $X\times Y$ is written $x\otimes y$. Now translate this whole train of thought to the category of $\mathbb{R}$-vector spaces. In this category, direct sum is the coproduct. if $X,Y$ are two $\mathbb{R}$-vector spaces, then their direct sum is written $X\oplus Y$. $X$ and $Y$ both have canonical maps to $X\oplus Y$. The element of $X\oplus Y$ the OP wants a name for is the sum of the canonical images of $x\in X$ and $y\in Y$ in $X\oplus Y$. In strict analogy with the tensor product case, why not call it $x\oplus y$?)
