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Homework Help: Linear Functionals

  1. Dec 10, 2006 #1
    I am studying for a final I have tomorrow in linear algebra, and I am still having trouble understanding linear functionals. Can someone help me out with this example problem, walk me through it so I can understand exactly what a linear functional is?

    Is the following a linear functional?

    [tex]\ y (x)=\int_0^1\ t^2 x(t) \, dx [/tex]
    [tex]\ y (x)=x(-2)+\int_0^1\ x(t^2)\, dt [/tex]
     
  2. jcsd
  3. Dec 10, 2006 #2

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    Always start by going back to the definitions.
     
  4. Dec 10, 2006 #3
    For the second one, which is basically just an addition to the first, is this correct?

    Property 1 of a linear functional is satisfied as follows:
    [tex]\ y (x+z)=x(-2)+z(-2)+\int_0^1\ (x(t^2)+z(t^2))\, dt [/tex]
    [tex]\ y (x+z)=x(-2)+\int_0^1\ x(t^2)\, dt + z(-2)+\int_0^1\ z(t^2)\, dt [/tex]
    [tex]\ y (x+z)=y(x)+y(z) [/tex]

    Property 2 of a linear function is satisfied similarly:
    [tex]\ y(a x)=a x(-2)+\int_0^1\ a x(t^2)\, dt [/tex]
    [tex]\ y(a x)=a (x(-2)+\int_0^1\ x(t^2)\, dt) [/tex]
    [tex]\ y(a x)=a y(x) [/tex]
     
    Last edited: Dec 10, 2006
  5. Dec 10, 2006 #4

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    It's as easy as that.
     
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