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 $\mathbb{R}^3$?
 A: It means to bring a two dimensional space to a three dimensional space, i.e., mapping vectors of the form $(x,y)$ to vectors of the form $(x,y,z).$ This must be done by applying some kind of linear transformation.
A: Here's an example. Let's take a vector from $\mathbb{R}^2\rightarrow\mathbb{R}^3$. The left hand side has the (transformation)(2D vector) = (3D vector) on the right side.
$$\begin{pmatrix} a & b\\c & d\\e & f\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix} ax+ by\\cx + dy\\ex + fy\end{pmatrix}$$

What's a real space $\mathbb{R}^n$ of dimension $n$? Let's explore this by visualizing some familiar spaces:


*

*A line is a 1D space, $\mathbb{R}$. You need only one coordinate $x$ to identify where you are. In this case, you don't need a vector... Since we only need one number, we have a scalar.

*A plane is 2D space, $\mathbb{R}^2$. You need two coordinates $(x,y)$ to locate a point. Instead of using an ordered pair, we can use a vector $\begin{pmatrix}x\\y\end{pmatrix}$.

*The world we live in is $\mathbb{R}^3$ from Euclid's point of view. As a vector a location can be specified by $\begin{pmatrix}x\\y\\z\end{pmatrix}$. Note that the number of entries in the column denotes the dimension.

*Higher dimensional spaces extend this idea. For an $n$-dimensional space, you can represent a point's location y a $n \times 1$ vector $\begin{pmatrix}x^{(1)}\\x^{(2)}\\x^{(3)}\\\vdots\\x^{(n)}\end{pmatrix}$.
The example above multiples a 2D vector on the left by some $3\times 2$ transformation matrix. The result is a 3D vector.
A: 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 
$$ \begin{bmatrix}7 \\ 15 \\ 23 \end{bmatrix} \in \mathbb{R}^{3} $$
