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Homework Help: Finding a basis of a Kernel

  1. Mar 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Let V = C(R,R) be the vector space of all functions f : R −> R that have continuous
    derivatives of all orders. We consider the mapping T : V −> V defined for all u belonging to V , by T(u(x)) = u''(x) + u'(x) − 2u(x). (Where u' is first derivative, u'' second derivative)

    Question: Determine Ker{T} and find a bsis of Ker{T}

    2. The attempt at a solution

    So as far as I understand the Ker{T} is the set of elements in T that maps into
    the zero vector, i.e. Ker{T} is when T(u(x))=0 (is this correct?)

    So T(u(x)) = 0 only when u''(x) + u'(x) = 2u(x). So would the Kernel of T be all functions
    u belonging to V such that u''(x) + u'(x) = 2u(x)

    But in this case how can I compute the basis?




    ---
    Thank you very much.
    Best Regards,
    SimpleDude
     
  2. jcsd
  3. Mar 5, 2008 #2
    solve for u and show the solutions are linearly independent
     
    Last edited: Mar 5, 2008
  4. Mar 5, 2008 #3
    Ok, so u(x) = [u''(x) - u'(x)] / 2

    But that's only one solution, how can I get anoter one from the same equation?
    (because if I use the same equaton, they both will be linearly dependant).

    Any help will be appreciated -- thanks!
     
  5. Mar 5, 2008 #4
  6. Mar 5, 2008 #5
    Ok I understand, and as far as the initial conditions go, if T is a linear operator
    is it ok to say that u(0)=0 and u'(0)=0 ?

    Because in that case everythign is 0 (both constants are 0, hence u(x)=0 )
     
    Last edited: Mar 5, 2008
  7. Mar 5, 2008 #6
    no you won't have initial conditions, so if you solved it(i'm assuming you did), you should get that u is of the form,

    u(x) = ae^(x) + be^(-2x) for some a, b in R, in particular, this means that

    u belongs to span({e^(x), e^(-2x)} = B(I just called it B for notation), so now you've shown
    that kerT is contained in spanB, and the other inclusion is obvious(you should check it to be sure), so kerT = spanB

    Now just show it's independent, this might be trickier for you, so i'll get you started. We'll just use the definition of linear independence.

    So suppose ae^x + be^(-2x) = 0 for some scalars a, b, in R.

    We need to show a = b = 0.

    hint: this equation must hold for ALL values of x, so try some different values and try to force a = b = 0.
     
  8. Mar 5, 2008 #7
    Ahhh :)

    Can I create a matrix of coefficients of
    u(x) and u'(x), namely:

    u(x) = ae^-2x + be^x
    u'(x) = -2ae^-2x + be^x

    And solve AX=0 via gauss?

    Doing so I get a matrix of the form
    1 1
    0 1

    Hence the are linearly independant, furhermore (1,0) and (1,1) form a basis of Ker T.
     
  9. Mar 5, 2008 #8
    The B I wrote above will be your basis, remember B is a subset of V which is a space of functions, (1, 0) and (1, 1) are elements of R^2, they certainly aren't functions!

    I gave you a hint, it's a good one, try it!:)


    Edit: Another way is to compute the wronskian of e^(-2x) and e^x and notice it's nonzero, but imo it's better to do it the way I suggested. What I suggested is a good trick to keep around because it's useful in other places(like when working with dual spaces, etc).
     
    Last edited: Mar 5, 2008
  10. Mar 5, 2008 #9
    Ok, that's not that hard.

    Since: e^x != e^-2x for any values of x, except 0, and e^0 = 1.
    The only way to satisfy the equation ae^x + be^x = 0
    is if th coefficients a,b=0 i.e. a=b=0

    Hence it is linearly independat and the basis of Ker T is indeed {e^-2x, e^x}
     
    Last edited: Mar 5, 2008
  11. Mar 5, 2008 #10
    that's not enough, the way you wrote it isn't really correct, but the idea is correct!

    We have ae^x + be^(-2x) = 0, we need to show a = b = 0. This equation holds for all values of x. So plug some values of x in.

    Set x = 0. Then we get a + b = 0, so b = -a. Now go back to the original equation,

    ae^x + be^(-2x) = ae^x - ae^(-2x) = a(e^x - e^(-2x)) = 0. Then a = 0, otherwise e^x - e^(-2x) = 0, that is e^x = e^(-2x) and this holds for all x in R, a contradiction(for example setting x = 1 shows it's clearly false).

    Therefore b = -a = -0 = 0, so a = b = 0 and we have independence, and B is a basis as needed.


    EDIT: I edited my post to give you more since I noticed you were really close.

    I hope this helps, it's a good trick to keep around.
     
    Last edited: Mar 5, 2008
  12. Mar 5, 2008 #11
    Ok so more abstratly,

    Let's take an arbitrarily large x, x --> infinity
    The equation e^x will go to infinity
    The equation e^-2x will go to zero.

    So a(infinity) + b(0) = 0 is only satisfied if a=0

    Now lets take x --> -(infinity)
    Th equation e^x will go to zero.
    The equation e^-2x will go to infinity.

    So a(0) + b(infinitiy)=0 is only satisfied if b=0

    Hence a=b=0
     
  13. Mar 5, 2008 #12
    hehe thanks!
    However, is my approach correct?
    Can infinity be used?


    Also Dan, if you don't mind me asking a more abstract question.
    Let's say I wanted to show that T was a linear operator.
    Can I independantly show each part of the operator? I mean can I show
    u''(x) is linear, then u'(x) is linear, and -2u(x) is linear.

    Then since they are all linear, a linear combination u''(x)+u'(x)-2u(x)
    is also linear, hence T is a linear operator.
     
    Last edited: Mar 5, 2008
  14. Mar 5, 2008 #13
    hmm no not really to the infinity thing, for practice you could show that {e^(bx), e^(cx)} is linearly independent if c != b. (using the same method I described)

    As for showing T is linear, yes, exactly what you said! It's actually easier, from calculus you know that differentiation is a linear operator, ie, d/dx(u + v) = du/dx + dv/dx, d/dx(cu) = c*du/dx, and the same when you take the second derivative because the composition of linear operators is linear, so we had

    T(u) = u'' + u' - 2u.

    So from calculus, it's immediate that
    T(u + v) = (u + v)'' + (u + v)' - 2(u + v)
    = u'' + v'' + u' + v' -2u - 2v
    = u'' + u' - 2u + v'' + v' - 2v
    = T(u) + T(v)

    and similarly T(cu) = cT(u)

    where we used the fact that differentiation is a linear operator
     
  15. Mar 5, 2008 #14
    Thank you sir!
     
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