TITLE: What does $\mathbb{R}^n \to \mathbb{R}^m$ mean? And what is $\mathbb{R}^n$?
QUESTION [11 upvotes]: What the does $\mathbb{R^n}$ mean? For example if something says that it is a transformation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Does that mean that $\mathbb{R}^2 = 2 \times 2$ matrix? and that $\mathbb{R}^3 = 3 \times 3 $ matrix?

REPLY [6 votes]: The symbol $\Bbb R^n$ refers to $n$-dimensional Euclidean space. As a set, it is the collection of all $n$-tuples of real numbers. That is,
$$
\Bbb R^n=\{(x_1,\dotsc,x_n):x_1,\dotsc,x_n\in\Bbb R\}
$$
For example $\Bbb R^2$ is the collection of all pairs of real numbers $(x,y)$, sometimes referred to as the Euclidean plane. The set $\Bbb R^3$ is the collection of all triples of numbers $(x,y,z)$, sometimes referred to as $3$-space.
Now, it is a fact that every linear transformation $T:\Bbb R^n\to\Bbb R^m$ is of the form $T(x)=Ax$ for some $m\times n$ matrix $A$.
In general, a function $F:\Bbb R^n\to\Bbb R^m$ is of the form
$$
F(x_1,\dotsc,x_n)=\bigl(f_1(x_1,\dotsc,x_n),\dotsc,f_m(x_1,\dotsc,x_n)\bigr)
$$
where $f_1,\dotsc,f_m$ are functions $\Bbb R^n\to\Bbb R$.