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Notation question - R^m -> r^n

  1. Apr 22, 2013 #1
    Notation question - R^m ---> r^n

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

    [itex]R^{m} \rightarrow R^{n}[/itex]

    I can't find the place in my book where it explains it.
     
  2. jcsd
  3. Apr 22, 2013 #2

    dx

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    It means a function from the space Rm to the space Rn.

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

    {(x1, ..., xm) | xi in ℝ}
     
  4. Apr 22, 2013 #3
    Hm, so you mean turning a vector in R3 to R4 as a concrete example?
     
  5. Apr 22, 2013 #4

    dx

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    Yep.
     
  6. Apr 22, 2013 #5
    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 [itex]f:A\to B[/itex] is denoted [itex]\Gamma_f[/itex] and is a subset of the product space [itex]A\times B[/itex].

    Linear algebra considers the case that f is linear.
     
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