11
$\begingroup$

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?

$\endgroup$
5
  • 2
    $\begingroup$ 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 $R^{3\times 2}$. $\endgroup$ Jul 7 '16 at 19:32
  • $\begingroup$ 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 $\mathbb{R}^3$) ? $\endgroup$
    – almagest
    Jul 7 '16 at 19:33
  • $\begingroup$ $\mathbb{R}^n$ 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. $\endgroup$
    – JasonM
    Jul 7 '16 at 19:33
  • 1
    $\begingroup$ @XianjinYang : ​ "The domain of $T$ is" $\mathbb{R}^2$ itself, not "a vector in $R^2$". ​ ​ ​ ​ $\endgroup$
    – user57159
    Jul 7 '16 at 19:36
  • $\begingroup$ @RickyDemer Thanks for the correction. I mean the elements in the domain. $\endgroup$ Jul 7 '16 at 19:40
16
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ So if it said $\mathbb{R}^3$ then it would be a $3 \times 1 $ matrix right? $\endgroup$
    – Yusha
    Jul 7 '16 at 19:42
  • 1
    $\begingroup$ @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. $\endgroup$
    – Thomas
    Jul 7 '16 at 19:42
6
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.