# What does $\mathbb{R}^n \to \mathbb{R}^m$ mean? And what is $\mathbb{R}^n$?

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?

• No. The domain of $T$ is a vector in $R^2$ and the range of $T$ is a vector in $R^3$. $T$ can be viewed as a matrix in $R^{3\times 2}$. Jul 7 '16 at 19:32
• You say you don't understand what $\mathbb{R}^n$ means. Do you have difficulty with the real plane (ie $\mathbb{R}^2$) or real 3D space (ie $\mathbb{R}^3$) ? Jul 7 '16 at 19:33
• $\mathbb{R}^n$ is the set of all points in $n$-dimensional space. $\mathbb{R}^2$ is not a $2\times2$ matrix, but the transformation $T$ can be represented by a $3\times 2$ matrix. Jul 7 '16 at 19:33
• @XianjinYang : ​ "The domain of $T$ is" $\mathbb{R}^2$ itself, not "a vector in $R^2$". ​ ​ ​ ​
– user57159
Jul 7 '16 at 19:36
• @RickyDemer Thanks for the correction. I mean the elements in the domain. Jul 7 '16 at 19:40

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

1. $T(v + w) = T(v) + T(w)$
2. $T(av) = aT(v)$

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 $Av\in \mathbb{R}^m$.

• So if it said $\mathbb{R}^3$ then it would be a $3 \times 1$ matrix right? Jul 7 '16 at 19:42
• @Yusha: You can also view $\mathbb{R}^n$ as $n\times 1$ matrices. That is right. I would usually want to think about them as vectors. Jul 7 '16 at 19:42

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 $\Bbb R^n\to\Bbb R$.