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Linear transformation T: R3 -> R2

  1. Dec 26, 2011 #1
    Linear transformation T: R3 --> R2

    1. The problem statement, all variables and given/known data

    Find the linear transformation T: R3 --> R2 such that:
    T(1,0,0) = (2,1)
    T(0,1,1) = (3,2)
    T(1,1,0) = (1,4)



    3. The attempt at a solution

    I've been doing some exercises about linear transformations (rotations and reflections mostly) but I've never seen something like this... I don't know how to even start :S I did some research and only found examples on proving the two conditions for a transformation to be linear...

    Any help please?
     
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  3. Dec 26, 2011 #2

    Dick

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    Re: Linear transformation T: R3 --> R2

    Could you answer the problem if they gave you the values of T(1,0,0), T(0,1,0) and T(0,0,1)? If you can than can you figure out what they are from the given information?
     
    Last edited: Dec 26, 2011
  4. Dec 26, 2011 #3
    Re: Linear transformation T: R3 --> R2

    Thanks for the reply.

    Humm.. what do you mean? If they changed the values of the T's I posted?
     
  5. Dec 26, 2011 #4

    Dick

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    Re: Linear transformation T: R3 --> R2

    Yes, T is a 2x3 matrix. T(1,0,0) is the first column of the matrix, right?
     
  6. Dec 26, 2011 #5
    Re: Linear transformation T: R3 --> R2

    Yeah, I would say so... Like I said, I have never done an exercise like this before, so I don't exactly now how to start arranging things hehe
     
  7. Dec 26, 2011 #6

    Dick

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    Re: Linear transformation T: R3 --> R2

    Well, it's time to start arranging things now. You know T(1,0,0). They gave it to you. So you know the first column of the matrix. Now what's T(0,1,0)? That would be the second column. Can you figure that out using linearity from the given information?
     
  8. Dec 27, 2011 #7
    Re: Linear transformation T: R3 --> R2


    hehe ok! Are you mistaking that second T? It should be T(0,1,1), right? Anyway, what exactly are we trying to do? Arrange T's inside a matrix as columns and then...?
     
  9. Dec 27, 2011 #8

    HallsofIvy

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    Re: Linear transformation T: R3 --> R2

    Another way to do this: Write <x, y, z> as a combination of <1, 0, 0>, <0, 1, 1>, and <1, 1, 0>. That is, <x, y, z>= a<1, 0, 0>+ b<0, 1, 1>+ c<1, 1, 0>= <a+ c, b+ c, b>. So we have a+ c= x, b+ c= y, and b= z. Then c= y- z and a= x- y+ z.

    That is, <x, y, z>= (x-y+ z)<1, 0, 0>+ z<0, 1, 1>+ (y- z)<1, 1, 0>.

    So T<x, y, z>= (x- y+ z)T<1, 0, 0>+ zT<0, 1, 1>+ (y- z)T<1, 1, 0>.
     
  10. Dec 27, 2011 #9

    Dick

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    Re: Linear transformation T: R3 --> R2

    What is T(1,1,0)-T(1,0,0)? In the example they gave it is easy to find the value of T on the three basis vectors (1,0,0), (0,1,0) and (0,0,1). That will let you write down a matrix for T. I was guessing that was what they meant by 'find the linear transformation'.
     
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