Results 1 to 3 of 3

Math Help - Show that derivation is linear?

  1. #1
    Member
    Joined
    May 2008
    Posts
    87

    Show that derivation is linear?

    I'm completely stuck on this one:

    "Derivation Dp = p' defines a transformation D : P_2 \rightarrow P_2. Use the definition of linear maps and the properties of derivation to show that D is linear. Describe the codomain and the range of D. What is the kernel of D? What are the dimension of range(D) and kernel(D)? Is the map onto? Is it one-to-one? Is it an isomorphism?"

    Where should I start? What should I read in order to understand this? Is this a problem that is understood simply through general experience with calculus and linear algebra?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by posix_memalign View Post
    I'm completely stuck on this one:

    "Derivation Dp = p' defines a transformation D : P_2 \rightarrow P_2. Use the definition of linear maps and the properties of derivation to show that D is linear. Describe the codomain and the range of D. What is the kernel of D? What are the dimension of range(D) and kernel(D)? Is the map onto? Is it one-to-one? Is it an isomorphism?"

    Where should I start? What should I read in order to understand this? Is this a problem that is understood simply through general experience with calculus and linear algebra?
    Which part don't you understand? the fact that the derivative defines a linear transformation is a trivial fact from calculus. So, for the kernel, you tell me? In general one of the guys in \mathcal{P}_2 looks like a+bx+cx^2 and so we want to solve this general guy for 0(x)=D(a+bx+cx^2)=b+cx. Recall then that a polynomial is zero, by definition, if and only if it has all zero coefficients Lastly, what do you think the image is? Let me ask you, for example, does there exists p\in\mathcal{P}_2 such that p'(x)=x^2?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by posix_memalign View Post
    I'm completely stuck on this one:

    "Derivation Dp = p' defines a transformation D : P_2 \rightarrow P_2. Use the definition of linear maps and the properties of derivation to show that D is linear. Describe the codomain and the range of D. What is the kernel of D? What are the dimension of range(D) and kernel(D)? Is the map onto? Is it one-to-one? Is it an isomorphism?"

    Where should I start? What should I read in order to understand this? Is this a problem that is understood simply through general experience with calculus and linear algebra?
    First note that a basis for

    \mathbb{P}_2=\{1,x,x^2\}=\{\mathbf{e}_1,\mathbf{e}  _2,\mathbf{e}_3\}

    Now using the definition of the derivative we get that

    T(\mathbf{e}_1)=D(1)=0
    T(\mathbf{e}_2)=D(x)=1=\mathbf{e}_1
    T(\mathbf{e}_3)=D(x^2)=2x=2\mathbf{e}_2

    So this transformation has the matrix representation

    \begin{bmatrix}0 & 1& 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{bmatrix}

    Does this help? see if you can finish from here!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivation and Jordan derivation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 8th 2011, 10:22 PM
  2. show that a vector is in the range of a linear transform
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 20th 2009, 11:21 PM
  3. Show set of linear transformations from R^n to R^m
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 31st 2009, 07:53 PM
  4. How to show this is Linear? (Urgent!)
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 29th 2008, 09:50 AM
  5. Replies: 1
    Last Post: April 8th 2008, 12:23 AM

Search Tags


/mathhelpforum @mathhelpforum