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Homework Help: Isomorphic transformation question

  1. Jun 21, 2011 #1
    there is a transformation T:R^4 ->R^4


    dim Im(T+I)=dim Ker(3I-T)=2

    prove that T-I is isomorphic

    first of all i couldnt understand the first equation

    because T is a transformation which is basicly a function

    but I is the identity matrices

    so its like adding kilograms to tempreture.

    then my prof told me that here I is a transformation too

    that its not a matrix

    its the identity transformation.

    and i was told that i need to get the eigenvectors from there

    so i told him

    "how i dont have any matrix here to do


    i dont have any matrices only some transformation

    which i dont have even the formula to T in order to get the representing matrices out if it

    how to find eigen vectors from here


    and how to proceed in order to prove that


    is isomorphic

  2. jcsd
  3. Jun 21, 2011 #2
    Hi nhrock3! :smile:

    Matrices and transformations are equivalent. Given a matrix, there is a corresponding transformation and vice versa. Thus we can define the notion of eigenvalue for a transformation by

    [tex]T(x)=\lambda x[/tex]

    thus the eigenvectors of T are the elements of

    [tex]Ker(T-\lambda I)[/tex]

    and lambda is the corresponding eigenvalue.

    Now, can you use the fact that

    [tex]dim Ker(T+I)=2[/tex]

    to find an eigenvalue of T? And what multiplicity does the eigenvalue have?
  4. Jun 21, 2011 #3
    i dont understand the transition
    i know that
    [tex]T(v)=Av=\lambda v[/tex]
    its the definition of the link between eigen vectors and eigen values

    Ker(T+I) means (T+I)(v)=0

    but T and I are both transformation
    so i cant use it like here
    in here
    [tex]Ker(T-\lambda I)[/tex]
    I is a mtrix
    but my I is the identity transformation
    its a function not a matrix
    Last edited: Jun 21, 2011
  5. Jun 21, 2011 #4
    ok i understand your idea
    Ker(T+I) meand the eigen vectors of eigenvalue -1
    but still matheticly
    i need to replace T with some matrices and I (the transformation) needs to be I(the identety matrix)
  6. Jun 21, 2011 #5
    Why do you change everything to matrices?? You can do that, of course, but you can leave everything in transformation form too.

    Saying that the vectors in


    are the eigenvectors of T with eigenvalue -1, is perfectly fine for transformations. There's no need to change everything to matrices. But you can change to matrices if it's easier for you...
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