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Math Help - Finding Linear Transformation

  1. #1
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    Finding Linear Transformation

    Hi,

    I'm having trouble with this subject and would like to ask for some help and explanations.

    For example, I have the following question:

    V1 = (1,1,1)
    V2 = (2,3,1)
    Find linear transformation T:R3 -> R3 such that:
    a. Its kernel spans by V1 and V2
    b. Its kernel span by V1 only

    How should I approach this?
    Any help will be appreciated.

    Thanks
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  2. #2
    MHF Contributor
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    Re: Finding Linear Transformation

    Hey MathStudent111111.

    Hint: If it only spans V1 then it will only have scalar multiples of V1.
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  3. #3
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    Re: Finding Linear Transformation

    Just recall that a linear transformation is completely determined if we know the image of the vectors in any base of the domain.

    So, first of all, find a third vector v_3 so that \{v_1,v_2,v_3\} is a base for \Bbb R^3.

    A. You want the kernel of T to be the span of \{v_1,v_2\}, therefore you need to set T(v_1)=(0,0,0) and T(v_2)=(0,0,0). Additionally, you have to make T(v_3) non-zero (this way you don't get a bigger kernel).

    B. Set T(v_1)=(0,0,0), and send v_2 and v_3 to two non-zero linearly independent vectors in \Bbb R^3, say, (1,0,0) and (0,1,0).
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