1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Homework Helper

    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


    User Avatar
    Homework Helper

    It's as easy as that.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook