Kernel and Range of a Linear Mapping

[Smazmbazm]

Homework Statement

Find the kernel and range of the following linear mapping.

b) The mapping T from [itex]P^{R} to P^{R}_{2}[/itex] defined by

[itex]T(p(x)) = p(2) + p(1)x + p(0)x^{2}[/itex]

The Attempt at a Solution

I'm not sure how to go about this one. Normally I would use the formula T(x) = A * v but in this case I don't know how to find A or v. Would be great if someone could point me in the right direction.

Thanks in advanced.

 

[HallsofIvy]

I assume you know that the kernel is the set mapped to 0. T(p(x))= 0 (for all x) if and only if p(2)= p(1)= p(0)= 0.

And the range is the entire set P^R₂. Do you see why?

 

[Smazmbazm]

Yes I know that the kernel is the set that is mapped to 0. I think I'm just having trouble with understanding what [itex]T(p(x)) = p(2) + p(1)x + p(0)x^{2}[/itex] actual means. It's a bit too general for me. What is [itex]p(x)[/itex]? Is that saying that the power representation for [itex]p(x)[/itex] is [itex]p(2) + p(1)x + p(0)x^{2} [/itex]? Or the mapping of the power series [itex]p(x)[/itex] from [itex]P^{R}[/itex] to [itex]P^{R}_{2}[/itex] results in [itex]p(2) + p(1)x + p(0)x{2}[/itex]

I think I understand why the range is the entire set [itex]P^{R}_{2}[/itex], because if the kernel only contains the zero vector then the range must contain everything else?