Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By TheEmptySet

Math Help - Find the basis of kernal and image of a transformation

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Find the basis of kernal and image of a transformation

    Suppose that n is a positive integer, and let the transformation T:C(x) \rightarrow C(x) be defined by T(P(x))=xp'(x)-np(x) \ \ \ \ \ p(x) \in C(x) where C(x) is the set of continuous function mapping from  \mathbb {N} to a set M.

    Find a basis of kernal(T) and Image(T)

    so far:

    So I know that we need to find the set  K = \{ p(x) \in C(x) : T(p(x)) = 0 \}

    Now, if  T(p(x)) = xp'(x)-np(x) = 0

    we will have  p'(x)= \frac {np(x)}{x}

    So I need to find the set of function that have that behavior?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Find the basis of kernal and image of a transformation

    Quote Originally Posted by tttcomrader View Post
    Suppose that n is a positive integer, and let the transformation T:C(x) \rightarrow C(x) be defined by T(P(x))=xp'(x)-np(x) \ \ \ \ \ p(x) \in C(x) where C(x) is the set of continuous function mapping from  \mathbb {N} to a set M.

    Find a basis of kernal(T) and Image(T)

    so far:

    So I know that we need to find the set  K = \{ p(x) \in C(x) : T(p(x)) = 0 \}

    Now, if  T(p(x)) = xp'(x)-np(x) = 0

    we will have  p'(x)= \frac {np(x)}{x}

    So I need to find the set of function that have that behavior?
    This is a separable ODE.

    \frac{dp}{dx}=\frac{np}{x} \iff \frac{dp}{p}=\frac{n}{x}

    Integrating both sides gives

    \ln|p|=n\ln|x|+c \iff p(x)=Ax^n
    Thanks from tttcomrader
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basis of Image & Kernal
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 20th 2012, 03:35 PM
  2. Replies: 1
    Last Post: April 11th 2012, 12:59 AM
  3. Replies: 1
    Last Post: September 30th 2011, 04:36 PM
  4. Replies: 3
    Last Post: February 11th 2009, 12:34 AM
  5. Kernal and Image of a Linear Transformation
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 15th 2008, 06:04 PM

Search Tags


/mathhelpforum @mathhelpforum