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Math Help - Maps / Rank and Nullity question

  1. #1
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    Maps / Rank and Nullity question

    Hey guys, I've been trying this question but never get the same result as the answer sheet. It doesn't go into much, if any detail at all about it's methods so I was wondering if anybody can help me with it.



    I know what rank and nullity are, and also the properties of linear mapping. But I just can't even figure out where to start.

    Please could someone give me some hints, or maybe even just a step by step.

    This is much appriciated - Thank you.
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  2. #2
    MHF Contributor

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    Do you know the definition of "Linear map"? The first thing you are asked to do is show that this is a linear map so show that the definition of "linear map" applies.
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  3. #3
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    I know how to show the map is linear, using (L1) and (L2) Criterion, Showing Q(v+w) = Q(v) + Q(w) and also Q(aV) = aQ(v).

    I am more just struggling with the rank and nullity, I know what they are but I just can't even figure out where to start here - My head may just be a bit jammed with all the work I've done recently.
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  4. #4
    Senior Member roninpro's Avatar
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    Recall that the kernel of \varphi is the set of vectors v\in V such that \varphi(x)=0. (Note that this forms a subspace.) And, the nullity of \varphi is the dimension of that kernel.

    You first should try to compute the kernel of \varphi; that is, find all of the vectors satisfying \varphi(ax^3+bx^2+cx+d)=(a+b+d,2a+b+c-d,3a+2b+c,2d-a-c)=(0,0,0,0). Then you can find a basis and get the dimension.
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  5. #5
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    Thank you, I have figured out where I was going wrong - It was because I was placing the (a,b,c,d) on the right hand side into a matrix, placing it into rREF but then thinking I was finished, rather than placing it into the solved version of the linear equations

    Thanks for the help both of you.
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