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Homework Help: Find dimension and ker of matrices ?

  1. Apr 14, 2012 #1
    Find dimension and ker of matrices ??

    Let V be an F-vector space and (phi:v->v) be an F-linear transformation of V . Define what
    it means for a vector v ε V to be an eigenvector of phi and what is meant by the associated
    eigenvalue.


    This is the form of the question during my calculations I need to calculate:


    Now I have from the eigenvalues:

    dim(ker()) of a matrix:
    1 4 -3
    0 0 0
    0 0 0

    and

    dim(ker()) of a matrix:
    1 0 -2/7
    0 1 -5/7
    0 0 0

    Then when the I add up the the dim's I will be able to tell if it is diagonalisable.

    Now in this case it is obvious, the dim(ker()) of one is 1 and another is 2. But I can't tell which one is which.

    What I mean is the 1st Matrix = 2 and the second Matrix = 1 or viceversa ?

    In this case in either way it will add upto 3 and as the original matrix was a 3x3 matrix it is diagonalisable right ?

    In essence I don't really understand what dim or ker do ...
     
  2. jcsd
  3. Apr 14, 2012 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Re: Find dimension and ker of matrices ??

    I have no clue what you are asking. The title is "Find dimension and ker of matrices" but your first question appears to be about eigenvalues. You have "dim(ker()) of a matrix:" followed by what look like row reduced matrices. What are the original matrices?

    You say "Then when the I add up the the dim's I will be able to tell if it is diagonalisable." Tell if what is diagonalisable?

    The dimension of the kernel of a matrix tells you whether or not 0 is an eigenvalue and the algebraic dimension but tells you nothing about any non-zero eigenvalues- and does NOT tell you if the matrix is diagonalizable. In particular a row-reduced matrix does not have the same eigenvalues or eigenvectors as the original matrix.
     
  4. Apr 14, 2012 #3
    Re: Find dimension and ker of matrices ??

    My bad, I shouldn't have given the question in the first place as I just want to know what the resective dims kers are.

    1)
    what is the dim(ker()) of:
    1 4 -3
    0 0 0
    0 0 0

    2)
    and what is the dim(ker()) of:
    1 0 -2/7
    0 1 -5/7
    0 0 0

    What are the answers to 1) and 2) is all I want to know, also the dims of the above matrices will also be useful.

    I know that the answers are 1 and 2, but I'm not sure which one is 1 and which one is 2.

    If I am coming across confused, its because I am...sorry
     
  5. Apr 14, 2012 #4
    Re: Find dimension and ker of matrices ??

    The ranks of both matrices are obvious (just look at the columns). The nullities follow immediately with the rank-nullity theorem.
     
  6. Apr 14, 2012 #5
    Re: Find dimension and ker of matrices ??

    So for the first one the dim(ker()) is 3-1=2
    and for the second one the dim(ker()) is 3-2=1

    where 3 is # of columns and 1 and 2 are the ranks respectively ?
     
  7. Apr 14, 2012 #6
    Re: Find dimension and ker of matrices ??

    Yes, that's correct.
     
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