Notation question - R^m -> r^n

[Astrum]

Notation question - R^m ---> r^n

I've come across this notation a lot lately. I'm not sure what it really means.

$R^{m} \rightarrow R^{n}$

I can't find the place in my book where it explains it.

 

[dx]

It means a function from the space R^m to the space Rⁿ.

R^m is the m dimensional real vector space, or sometimes more concretely the space

{(x¹, ..., x^m) | xⁱ in ℝ}

 

[Astrum]

Hm, so you mean turning a vector in R³ to R⁴ as a concrete example?

 

[dx]

Yep.

 

[algebrat]

We can write
\begin{align}
f:\mathbb{R}^m&\to\mathbb{R}^n\\
x&\mapsto y=f(x)
\end{align}
and
\begin{align}
y&=(y_1,\dots,y_n)\\
&=f(x)\\
&=f(x_1,\dots,x_m)\\
&=(f_1(x),\dots,f_n(x))\\
&=(f_1(x_1,\dots,x_m),\dots,f_n(x_1,\dots,x_m)).
\end{align}
Try drawing the graphs of the functions
\begin{align}
f:\mathbb{R}^2&\to\mathbb{R}\\
(x,y)&\mapsto z=\sqrt{x^2+y^2}
\end{align}
and
\begin{align}
f:\mathbb{R}&\to\mathbb{R}^2\\
x&\mapsto (y,z)=(\cos x,\sin x).
\end{align}
The graph associated with the map $f:A\to B$ is denoted $\Gamma_f$ and is a subset of the product space $A\times B$.

Linear algebra considers the case that f is linear.