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Math Help - Find the Kernel of the linear transformations

  1. #1
    Jav
    Jav is offline
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    Question Find the Kernel of the linear transformations

    I've noticed that most of my problems come, when I try to solve spaces of polynomials rather than real numbers.
    Please, help.

    T: P3(all polynomials of third degree or less)-->R, T(a+bx+cx^2+dx^3)=a
    Find the kernel of the linear transformation.

    My approach starts as follows:
    According to my book, Elementary Linear Algebra by Larson and Falvo, the kernel of linear transformations is found by equating the right hand side of the T function to 0.
    Then, a=0. But, I can't make anything out of it.

    Most of the examples are very straight forward and in R. There are no examples on P spaces.
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  2. #2
    MHF Contributor

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    Re: Find the Kernel of the linear transformations

    Yes, that's exactly right. The "kernel" of a linear transformation is the set (a subspace) of vectors that the linear transformation takes to 0. If T(a+ bx+ cx^2+ dx^3)= a, then a+ bx+ cx^2+ dx^3 is in the kernel of T if and only if a= 0. That is the three dimensional subspace of all polynomials of the form bx+ cx^2+ dx^3. The kernel is {{P(x) | P(x)= bx+ cx^2+ dx^3}.

    Thanks, Deveno. I have no idea now why I wrote "four"!
    Last edited by HallsofIvy; July 6th 2011 at 06:53 AM.
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  3. #3
    MHF Contributor

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    Re: Find the Kernel of the linear transformations

    three-dimensional subspace. a four-dimensional subspace would be all of P3 (added just to clarify for the OP, i'm sure it was a typo on your part).
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