What does $\mathbb{R}^n \to \mathbb{R}^m$ mean? And what is $\mathbb{R}^n$?
Solution 1:
No, $\mathbb{R}^2$ means the space of $2$ dimensional vectors. For example $$ \pmatrix{7 \\ -2} $$ is an example of an element in $\mathbb{R}^2$.
More generally $\mathbb{R}^n$ means the space of all $n$-dimensional vectors. So, these are vectors have have $n$ coordinates.
The key thing is that $\mathbb{R}^n$ is a vector space. All this means is that you have an addition of the vectors and you have a scalar multiplication.
Now, you might also view $\mathbb{R}^n$ as points in a space. But it looks like you are thinking about $\mathbb{R}^n$ as vector spaces since you talk about linear transformations. A linear transformation $T$ between two vector spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, written $T: \mathbb{R}^n \to \mathbb{R}^m$ just means that $T$ is a function that takes as input $n$-dimensional vectors and gives you $m$-dimensional vectors. The function needs to satisfy certain properties to be a linear transformation. These properties are
- $T(v + w) = T(v) + T(w)$
- $T(av) = aT(v)$
for all $v,w\in \mathbb{R}^n$ and $a$ a real number.
When you have a linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$, then you can find a unique matrix $A$ such that $$ T(v) = Av $$ Here, $A$ is an $m\times n$ matrix. We need this for the product $Av$ to make sense and to get $Av\in \mathbb{R}^m$.
Solution 2:
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 $\Bbb R^n\to\Bbb R$.