Kernel and images of linear operator, examples

  • Thread starter Niles
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  • #1
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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.
 

Answers and Replies

  • #2
HallsofIvy
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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 [itex]p(x)= ax^2+ bx+ c[/itex], then [itex]p'(x)= 2ax+ b[/itex] and so L(p(x))= [itex]2ax^2+ bx[/itex]. That set of functions is the range. Since a can be any number, so 2a can be any number.
 
  • #3
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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.
 

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