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Finding basis for nullspace of transformation

  1. Oct 28, 2012 #1
    T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2.
    T (p(x)) = p(0)

    Find a basis for nullspace of linear transformation T.


    The answer is {x, x^2}

    I want to make sure I'm interpreting this correctly.

    It only goes to x^2 because of P2 right? Like if it was P3 it would be x, x^2, x^3?

    Also I don't understand why it's {x, x^2} is it because x can be 0?
     
  2. jcsd
  3. Oct 28, 2012 #2

    Mark44

    Staff: Mentor

    If p(x) = a2x2 + a1x + a0, what is T(p(x))?
     
  4. Oct 29, 2012 #3
    oooo, just a0. Which must be 0 if the p(x) you give is to satisfy the requimrenets for N(T) right?

    So essentially the problem is saying P(x)=x and P(x)=x^2?
     
  5. Oct 29, 2012 #4

    Mark44

    Staff: Mentor

    No. a0 can be any real number.
    No. How can P(x) be both x and x2?

    P(x) is an arbitrary 2nd degree polynomial.

    What does it mean for a function to be in the nullspace in this problem?

    What is a basis for P2? What does the transformation T do to each of these basis elements?
     
  6. Oct 29, 2012 #5
    It means it can be spanned by the vectors in the basis which is x and x^2.


    I think a basis for P2 is {1, x, x^2)?

    In my textbookthey define N(T)]{ all v in vector space V and T(v)=0.}
    So for this problem
    N(T)={P(x) such that P(0)=0}
    1 would be dropped out because P(0)=0 wouldn't be satisfied.
     
  7. Oct 29, 2012 #6

    Mark44

    Staff: Mentor

    Yes, but the only reason you know that the x and x2 are in a basis for the nullspace of T is because you are given the answer.
    Yes.
    No. N(T) = {P(x) ##\in## P2 such that T(P(x)) = P(0)}
    What does the transformation T do to each of the three functions in your basis?
     
  8. Oct 29, 2012 #7
    That's where I get a bit confused.

    T(1)=1
    P(x)=x then p(0)=0
    So
    T(x)=0
    T(x^2)=0
    ?
     
  9. Oct 29, 2012 #8

    Mark44

    Staff: Mentor

    What you have above is OK - where are you confused? You now have enough information to give a basis for Null(T).
     
  10. Oct 29, 2012 #9
    Got it, just learned how to do this today in class.
    Since
    T(1)=1
    T(x)=0
    T(x^2)=0

    They form the transformation matrix:
    1 0 0
    0 0 0
    0 0 0
    this is spanned by the vectors {[0 1 0]^t, [0 0 1]^t}
    Which correspond to x, and x^2
     
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