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Isomorphic transformation question

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

    so

    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

    |A-labdaI|=0

    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

    T-I

    is isomorphic

    ?
     
  2. jcsd
  3. Jun 21, 2011 #2

    micromass

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    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
    because
    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
    that
    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

    micromass

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    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

    [tex]Ker(T+I)[/tex]

    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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