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Homework Help: A question in lenear algebra

  1. Jan 17, 2008 #1
    in the link i wrote the question and the way i tried to solve it

    but i got a realy weird answer in the first part

    and in the second part i solved it too
    but i am not sure if my theoretical knowledge about finding ker and Im are right

    Last edited: Jan 17, 2008
  2. jcsd
  3. Jan 17, 2008 #2
    #1: The dimension of the nullspace of the matrix (which is the dimension of the kernel of the transformation that the matrix represents) is be the number of free variables. So 3a+b-c must equal zero.

    #2: Remember that the dim ker(T) + dim Im(T) = dim U, where dim U = dim R^4 = 4. Also keep in mind that dim Im(T) is the rank of the matrix that represents T.

    When the dimension of the kernel is minimum, the dimension of the image (the rank of A) is maximum and vice versa.
    Last edited: Jan 17, 2008
  4. Jan 17, 2008 #3
    i know all of this
    but in normal matrices thats how i would find the ker(t)
    what the problem with that??

    and about the second part did i make it right??
  5. Jan 17, 2008 #4


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    You find the kernel of T by solving the equation Tx= 0. In terms of a matrix with "unknowns" such as a, b, c, that means row reducing the matrix as much as you can, then choosing a, b, c so as to make rows all 0. The "maximal kernel" is, of course, the kernel of highest dimension. You find the maximal kernel by choosing a, b, c to make as many rows as possible all 0's. The second part of the question asks you to find a, b, c so as to make the image "maximal"- i.e. as high a dimension as possible. Since nullity+ rank= dimension of target space, (nullity= dimension of kernel, rank= dimension of image), you make the range maximal by making the kernel minimal: choose a, b, c so as to make as few rows as possible all 0.
  6. Jan 17, 2008 #5
    that what i did
    but i get an expresion of

    in order to the line to be all zeros
    i need to say

    the problem with that solution is that in order to find the ker(T)
    we need to SOLVE those equations
    why arent we solving them??

    about the second part i dont want to use the law of dimention
    i solved it in a certain way
    is that ok what i did??
    Last edited: Jan 17, 2008
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