Kernal and Range of a Linear Transformation

  1. Let L:p2 >>> p3 be the linear transformation defined by L(p(t)) = t^2 p'(t).
    (a) Find a basis for and the dimension of ker(L).
    (b) Find a basis for and the dimension of range(L).

    The hint that I get is to begin by finding an explicit formula for L by determining
    L(at^2 + bt + c).
    Does this hint mean let p(t) = at^2 + bt + c?
    Then, I find that t^2 p'(t) = 2at^3 + bt^2.
    Then, I conclude that the basis for ker(L) = {1}.
    Is it right?
    Also, how to find range(L)?

  2. jcsd
  3. hunt_mat

    hunt_mat 1,621
    Homework Helper



    This will be zero when t=0 or t=-b/2a, so...
  4. Dick

    Dick 25,914
    Science Advisor
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    That doesn't have much to do with the problem. hkus10 correctly has ker(L)={1} and having written L(p(t))=2at^3+bt^2 the answer to the dimension and a basis of range(L) should be pretty obvious. Why isn't it hkus10? What's a basis for p3?
  5. is the basis for ker(L) {t, 1} and the basis for range(L) {t^3, t^2}?
  6. Dick

    Dick 25,914
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    Two steps forward, one step backward. Yes, that's a basis for range(L). But now your basis for ker(L) is wrong. I liked your ker(L)={1} a lot better. Why did you put t in? Is t in the kernel?
  7. Char. Limit

    Char. Limit 1,937
    Gold Member

    I believe you have a workable basis for range(L). However, I think your basis for ker(L) has too many entries.
  8. Since L(at^2 + bt + c) = 2at^3 + bt^2
    No matter what value of t and 1, 2a^3 + bt^2 should always give me 0 vector. So, I have a question why Ker(L) does not have t as a basis?
    Another question is dim Ker(L) + dim range(L) = dim (p3) by thm. since the dim range(L) = 2 and dim (p3) = 4, why dim ker(L) not equal to 2?
  9. Dick

    Dick 25,914
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    The rank nullity theorem tells you dim ker(L)+dim range(L)=dim(p2). Not dim(p3). dim(p2)=3. t is not in ker(L) because L(t) is not zero. L(t)=t^2. t^2 is not zero.
  10. Thanks
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