Finding Kernel and Range for Linear Transformation L(p(x)) = xp'(x) in P3

[loli12]

Hi, does anyone know how to figure out the kernal and range for this linear transformation from P3 into P3 : L(p(x)) = xp'(x)?
I thought ker(L)= {0} and range is P3. But the correct answer is ker (L) = P1, L(P3) = Span (x^2, x). Can someone explain to me how exactly do we fine the kernel and range for this? Thanks!

 

[matt grime]

Pn is the polys of degree n right? Take an arbitrary degree three poly:

a+bx+cx^2+dx^3

and apply L to it.

What has been killed and what is left?

 

[loli12]

Well, my book defined Pn to be polynomials of less than n degree.
if I apply the L to ax^2 + bx + c, then i get L(p(x)) = 2ax^2 + bx + 0
i remembered from one of the ques that my teacher did, she set all those coefficients equal to 0 to find the kernal, if i do the same thing, i get a=b=c=0, so, shouldn't i get ker(L)= {0}? I don't understnad how to get P1 for that. Can you explain in a more detailed way. Thanks a lot!

 

[shmoe]

Well, my book defined Pn to be polynomials of less than n degree.
if I apply the L to ax^2 + bx + c, then i get L(p(x)) = 2ax^2 + bx + 0
i remembered from one of the ques that my teacher did, she set all those coefficients equal to 0 to find the kernal, if i do the same thing, i get a=b=c=0,

For the kernel, you don't set the coefficients to zero, you set L(p(x)) to zero and try to find what p(x)'s will satisfy this. If L(ax^2+bx+c)=0, then 2ax^2 + bx + 0=0. What choices of a,b,c will satisfy this?

 

[HallsofIvy]

Well, my book defined Pn to be polynomials of less than n degree.
if I apply the L to ax^2 + bx + c, then i get L(p(x)) = 2ax^2 + bx + 0
i remembered from one of the ques that my teacher did, she set all those coefficients equal to 0 to find the kernal, if i do the same thing, i get a=b=c=0, so, shouldn't i get ker(L)= {0}? I don't understnad how to get P1 for that. Can you explain in a more detailed way. Thanks a lot!

If p= ax²+ bx+ c, then p'= 2ax+ b so Li(p)= 2ax²+ bx . Setting "all those coeficients equal to 0" gives 2ax= 0 and b= 0.
How did you get c=0?

What is P1?