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Linear Algebra

  1. Mar 22, 2006 #1
    Fpr the matrix find a basis for hte kernel and image of [itex} T_{A} [/itex] and find the rank and nullity of [itex] T_{A} [/itex]
    T is a linear transformation


    [tex] \left(\begin{array}{cccc} 2&1&-1&3 \\ 1&0&3&1 \\ 1&1&-4&2 \end{array} \right) [/tex]

    the kernel of T simply means to find the null space of A right?
    so when i row reduce i get
    [tex] \left(\begin{array}{cccc} 1&0&3&1 \\ 0&1&-7&0 \\ 0&0&0&0 \end{array} \right) [/tex]

    so do i simply find a 3x1 line matrix X such taht AX = 0
    The image means something to do iwth the solution... but there is no augmented form given here... is there??

    the basis of A will be the rank T right? Is base A = 2?? So the rank of T = 2?
    the dimensio of the kernel is T is nullity of T... but i need to find the basis for the kernel first
     
  2. jcsd
  3. Mar 22, 2006 #2

    0rthodontist

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    The _rank_ of A will be the rank of T. Otherwise you are correct, what you need to do is find all the vectors x such that Ax = 0, from which you can extract a basis.
     
  4. Mar 22, 2006 #3
    ok s oteh rank of kernel of T is 2

    what about he nullity, though?? SO basically it asks how many vectors are forming the basis of th kernel?
     
  5. Mar 22, 2006 #4

    0rthodontist

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    No, the _dimension_ of the kernel of T is 2. That _is_ the nullity.
     
  6. Mar 22, 2006 #5
    ok ok
    so the number of independant rows in A: is the rank of A and that is 2
    that is the tank of Ta yes?

    also the rank of the kernel is the number of linearly independant vecotrs that can be formed from AX = 0 right

    p.s. are you an orthodontist?
     
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