Linear Functionals

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]
 

StatusX

Homework Helper
2,564
1
Always start by going back to the definitions.
 
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:

StatusX

Homework Helper
2,564
1
It's as easy as that.
 

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