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Homework Help: Linear Transformation part 2

  1. Mar 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Let R2 => R2 be a linear transformation for which we know that:

    L(1,1) = (1,-2)
    L(-1,1) = (2,3)

    What is: L(-1,5) and L(a1,a2)?


    2. Relevant equations

    I dont know where to start. I tried writing (-1,5) as a linear combo of (1,1) and
    (-1,1), but that got me nowhere. Am i suppose to find a basis? How do i do that?

    3. The attempt at a solution

    (-1,5) = a(1,1) + b(-1,1)

    Transforming it into a 2 x 3 matrix and row reducing it, i get a = 2, b = 3.
    If i am on the right track, what do i do next?
     
  2. jcsd
  3. Mar 7, 2008 #2

    Dick

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    If (-1,5)=2*(1,1)+3*(-1,1), then L(-1,5)=2*L(1,1)+3*L(-1,1), right? That's what 'linear' is all about. Now do the same thing for (a1,a2).
     
  4. Mar 8, 2008 #3
    Ok i think i got part a!

    For L(a1,a2), i stuck at: L(a1,a2) = 2(L(a1,a2)) + 3(L(a1,a2))

    I was looking at some of examples, and i had no idea where they get the (a+b)/2 and (a-b)/2 terms from.
     
  5. Mar 8, 2008 #4

    Dick

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    Aren't you going to solve (a1,a2)=a*(1,1)+b*(-1,1) first for a and b, so you can do EXACTLY the same thing as you did for part a? Except this time a=2 and b=3 are changed. I think that's the pedagogical point.
     
    Last edited: Mar 8, 2008
  6. Mar 8, 2008 #5
    so this is what i did:

    [ 1 -1 a1
    1 1 a2 ]

    [ 1 -1 a1
    0 2 a2 ]

    which means b = a2/2, a = a1 + a2/2

    Then L(a1,a2) = (a1 + a2/2)( L(1,1)) + (a2/2)(L(-1,2))

    = (a1 + a2/2)(1,-2) + (a2/2)(2,3)

    = (a1 + a2/2, -2a1 - a2) + (a2, (3a2)/2)

    = (a1 + (3a2)/2), (-2a1 + a2/2)

    But, this is not the right answer. What did i do wrong?
     
  7. Mar 8, 2008 #6

    HallsofIvy

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    This is wrong. You subtracted the first row from the second so this should be
    [ 1 -1 a1
    0 2 a2-a1]
    Of course, what you are really saying is that a- b= a1 and a+ b= a2. That should be easy to solve without matrix methods.
     
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