What is the meaning of $ \mathbb{R}^n$ to $\mathbb{R}^{n+1}$? In linear algebra, what does it mean to go from $\mathbb{R}^1$ to $\mathbb{R}^2$ or $\mathbb{R}^2$ to $\mathbb{R}^3$?

Let $T$ be a mapping from $\mathbb{R}^{2} \to \mathbb{R}^{3}$ be denoted by the matrix: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$. 
Then, $T$ sends vectors in $\mathbb{R}^{2}$ to $ \mathbb{R}^{3}$, for example: $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}\begin{bmatrix}1 \\ 3\end{bmatrix} = \begin{bmatrix}7 \\ 15 \\ 23 \end{bmatrix}$$
i.e.
$$ \begin{bmatrix}1 \\ 3\end{bmatrix} \in \mathbb{R}^{2} $$
but 
$$ \begin{bmatrix}7 \\ 15 \\ 23 \end{bmatrix} \in \mathbb{R}^{3} $$
