I have a vector including two vector's elements. How do I simply represent a vector with elemental vectors.
Formally, I have three vectors $x, a=(a_i), b =(b_i)$ and $x=(a_1, a_2, a_3,...,a_m,b_1,b_2,...,b_n)$. Of course $m$ and $n$ is a dimensionality of $a$ and $b$.
I'd like to know a simple way to represent $x$. The most naive one might be $(a, b)$ but I think it does not represent $x$ because $(a,b)$ is a two dimensional vector as $(a,b) = ((a_1, a_2, a_3,...,a_m),(b_1,b_2,...,b_n))$.
$(a,b)$ can be interpreted as an element of a product of vector spaces. In your case $(a,b)\in\Bbb R^m\times \Bbb R^n \cong \Bbb R^{m+n}$. So it is ok and commonly understandable to write it like that. If you want to emphasisze that $m,n\neq 1$ you could also write $(\mathbf{a},\mathbf{b})$ for $\mathbf{a}\in\Bbb R^m$ and $\mathbf{b}\in\Bbb R^n$.
Actually we write a vector $\mathbf{x}\in\Bbb R^2$ as $\mathbf{x}=(x_1,x_2)$ because $\Bbb R^2 = \Bbb R \times \Bbb R$ (again a product of vector spaces).