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Homework Help: Subspace, Linear Algebra, C^n[a,b]

  1. Feb 25, 2010 #1
    Show that C^n[a,b] is a subspace of C[a,b] where C^n is the nth derivative.

    I know the set is non empty since f(x)=x exist; however, I don't know how to start either the multiplication or addition property of subspaces to confirm that C^n is a subspace.

    Thanks ahead of time for any help any of you may have.
     
  2. jcsd
  3. Feb 25, 2010 #2
    What's the derivative of the sum of two functions? What's the derivative of a function multiplied by a constant?
     
  4. Feb 25, 2010 #3
    alpha*n*f^(n-1)(x)
    n*(f+g)^(n-1)(x)=n*[f^(n-1)(x)+g^(n-1)(x)]=n*f^(n-1)(x)+n*g^(n-1)(x)

    That works for generalization all nth derivative functions?
     
  5. Feb 25, 2010 #4
    Your expressions don't make much sense. If I'm reading them right (and please, try to use Latex), you're saying that:

    [tex]\left(\alpha f\left(x\right)\right)^{n}=n\alpha\left(\alpha f\left(x\right)\right)^{n-1}[/tex]

    That doesn't make much sense.
     
  6. Feb 25, 2010 #5
    multiplication: [tex]\alpha[/tex]*n*[tex]f^{n-1}(x)[/tex]
    addition: n*[tex](f+g)^{n-1}[/tex] = n*[[tex]f^{n-1}(x)[/tex]+[tex]g^{n-1}(x)[/tex]]
    and then the n distributes.

    Does this generalize all nth derivatives?

    Is there away for me to import from Maple since I have Maple and Latex is to slow and cumbersome?
     
  7. Feb 25, 2010 #6
    Those expressions are wrong, given functions f,g and a constant c, their n-th derivatives are:

    [tex]\left(f+g\right)^{\left(n\right)}\left(x\right)=f^{\left(n\right)}\left(x\right)+g^{\left(n\right)}\left(x\right)[/tex]

    And:

    [tex]\left(cf\right)^{\left(n\right)}\left(x\right)=cf^{\left(n\right)}\left(x\right)[/tex]
     
  8. Feb 25, 2010 #7
    But when the derivative is taking, don't the functions need to be multiplied by n and then the derivative is n-1?
     
  9. Feb 25, 2010 #8
    You are confusing the derivative of a power with the n-th derivative.
     
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