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Calculating Dim(ran(T) & Dim(Ker(T)

  1. May 23, 2009 #1
    I have a problem.
    Calculate Dim(Ran(T)) if T is 1-to-1. Also calculate Dim(Ker(T)) if T is onto.
    How do you think I should do this?
     
  2. jcsd
  3. May 24, 2009 #2

    matt grime

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    There is nothing to calculate. You just need to think about it. A 1-1 map is an isomorphism onto its image for example - plus there are the standard results like the rank nullity theorem to help you.
     
  4. May 24, 2009 #3

    HallsofIvy

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    If T is a linear transformation from a vector space of dimension n to a vector space of dimension m, then dim(Ran(T))+ dim(Kernel(T))= n. That's the "rank-nullity" theorem matt grime mentioned. If T is "one-to-one", then it maps only the 0 vector to the 0 vector so dim(Kernel(T))= ? If T is "onto" what is dim(Ran(T)).

    dim(Ran(T)) is also called the "rank" of T and dim(Kernel(T)) is the "nullity" of T.
     
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