Kernel and images of linear operator, examples

[Niles]

Homework Statement

If I e.g. want to find the kernel and range of the linear opertor on P_3:

L(p(x)) = x*p'(x),

then we can write this as L(p'(x)) = x*(2ax+b). What, and why, is the kernel and range of this operator?

The Attempt at a Solution

The kernel must be the x's where L(p'(x)) = 0, so do I just solve the equation?

The range is the elements that span the polynomial, so it is span{(x^2,x)}?

Hope you can help.

 

[HallsofIvy]

Homework Statement

If I e.g. want to find the kernel and range of the linear opertor on P_3:

L(p(x)) = x*p'(x),

then we can write this as L(p'(x)) = x*(2ax+b). What, and why, is the kernel and range of this operator?

The Attempt at a Solution

The kernel must be the x's where L(p'(x)) = 0, so do I just solve the equation?

Well, L(p)= 0, not L(p'). You must solve xp'(x)= 0 for all x.

The range is the elements that span the polynomial, so it is span{(x^2,x)}?

If every member of P_3 can be written as p(x)= ax^2+ bx+ c, then p'(x)= 2ax+ b and so L(p(x))= 2ax^2+ bx. That set of functions is the range. Since a can be any number, so 2a can be any number.

 

[Niles]

The kernel:
2ax^2+bx = 0 <=> 2ax+b=0 - is this the kernel, or do I have to isolate x first?

The range:
Ok, so the range is just the set of functions we get from L(p(x))?

I know these are quite basic subjects in linear algebra, but it is explained so poorely in my book and they've only spent ½ page giving definitions - nothing else.