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Proving range of transformation

  1. Oct 6, 2014 #1
    1. The problem statement, all variables and given/known data
    I haven't learned kernel yet so if thats of use here I don't know it yet

    let ##T: \mathbb{R^3} \rightarrow \mathbb{R^2}## where ##T<x,y,z>=<2y,x+y+z>##


    prove that the range is ##\mathbb{R^2}##




    3. The attempt at a solution
    I know that T is not one-to-one, I checked that for a previous question. I did read if the range is ##\mathbb{R^n}## then T is onto. but I am not really sure how to prove the range is ##\mathbb{R^2}##
     
  2. jcsd
  3. Oct 6, 2014 #2

    RUber

    User Avatar
    Homework Helper

    A good way to do this is to choose any (a,b) in ##\mathbb{R}^2##. Since they are arbitrary points in the range, if you can find a general form for them which is in the domain (in this case ##\mathbb{R}^3##), then the range is the entirety of ##\mathbb{R}^2##.
     
  4. Oct 6, 2014 #3
    so it goes both way "if ##T## is onto ##\Leftarrow\Rightarrow## the range is ##\mathbb{R^n}##"

    thanks

    I think I was misunderstanding what range meant in this sense.
     
  5. Oct 7, 2014 #4

    Mark44

    Staff: Mentor

    Since T is a transformation to ##\mathbb{R^2}## what you said should refer to that space, not ##\mathbb{R^n}##.
     
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