Understanding Linear Functionals: Help Me w/ Example Problem!

[wurth_skidder_23]

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?

$$\ y (x)=\int_0^1\ t^2 x(t) \, dx$$
$$\ y (x)=x(-2)+\int_0^1\ x(t^2)\, dt$$

 

[StatusX]

Always start by going back to the definitions.

 

[wurth_skidder_23]

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:
$$\ y (x+z)=x(-2)+z(-2)+\int_0^1\ (x(t^2)+z(t^2))\, dt$$
$$\ y (x+z)=x(-2)+\int_0^1\ x(t^2)\, dt + z(-2)+\int_0^1\ z(t^2)\, dt$$
$$\ y (x+z)=y(x)+y(z)$$

Property 2 of a linear function is satisfied similarly:
$$\ y(a x)=a x(-2)+\int_0^1\ a x(t^2)\, dt$$
$$\ y(a x)=a (x(-2)+\int_0^1\ x(t^2)\, dt)$$
$$\ y(a x)=a y(x)$$

 

[StatusX]

It's as easy as that.