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Is there a Linear Transformation

  1. Jan 19, 2015 #1
    1. The problem statement, all variables and given/known data
    From Hoffman and Kunze:

    Is there a linear transformation T from [tex]R^3[/tex] to [tex]R^2[/tex] such that T(1,-1,1)=(1,0) and T(1,1,1)=(0,1)?


    2. Relevant equations
    [tex] T(c\alpha+\beta)=cT(\alpha)+T(\beta) [/tex]

    3. The attempt at a solution
    I don't really understand how to prove that there is a linear transformation with these coordinates. I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1). Then I don't really know where to go from here.
     
  2. jcsd
  3. Jan 19, 2015 #2

    Orodruin

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    Think about how this helps you along with the requirement that T is a linear transformation. For example, what does your requirement that having the transformation of another linearly independent vector in R3 give you?
     
  4. Jan 19, 2015 #3
    It would allow me to span R3 with these three vectors
     
  5. Jan 19, 2015 #4

    Orodruin

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    And thus the transformation of any vector in R3 can be written as ...
     
  6. Jan 19, 2015 #5
    ... can be written as [tex]T(c\alpha_i+\beta_i)?[/tex]
    where alpha and beta are vectors in R3
     
  7. Jan 19, 2015 #6

    Orodruin

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    Well, you need to define what your ##\alpha## and ##\beta## etc are. You have three vectors with which you can express any vector in R3 as a linear combination. What will be the resulting transformation of such a linear combination? Will it fulfil the requirements you have?
     
  8. Jan 19, 2015 #7
    The resulting combination will be a vector in R^2, I think it could be any vector in R^2?
     
  9. Jan 19, 2015 #8

    Orodruin

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    No, not if you already have fixed it for three vectors. Use the linear property of T! Let us say you have the vectors ##v_1, v_2, v_3## and have fixed ##T(v_i) = u_i## for ##i = 1,2,3##. Now you take a linear combination of those ##v = \sum_i c_i v_i##. What is ##T(v)##?
     
  10. Jan 19, 2015 #9
    [tex] T(v)=T(c*v_1+c*v_2+c*v_3)[/tex]
    Which I can then apply the definition of a linear transformation as defined before?
     
  11. Jan 19, 2015 #10

    Orodruin

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    Yes. So does this linear transformation fulfil the requirements?
     
  12. Jan 19, 2015 #11
    Yes!!! it does! Thank you so much. It makes much more sense now. I just have one other question, why do the vectors need to span all of R^3 in the first place? Is it so the transformation can go to any place in R^2?
     
  13. Jan 19, 2015 #12

    Orodruin

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    Unless you specify the transformation for any arbitrary vector in R3 you have not really found a linear transformation from R3 to R2 but only from a two-dimensional subspace. A more complete answer would be: Yes, it exists, but is not unique. (You could have selected any vector in R2 as the image of your third linearly independent vector in R3).
     
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