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Linear Transformation

  1. Oct 19, 2005 #1
    I have a question regarding a math problem that I do not know how to go about solving.

    Let L: R^2 ---> R^2 be a linear operator. If L((1,2)^T)) = (-2,3)T
    and L((-1,1)) = (5,2)^T determine the value of L((7,5))^T

    Any insight would be much appreciated.
     
  2. jcsd
  3. Oct 19, 2005 #2
    L((1,2)^T)) = (-2,3)T

    Just to make things clear, L is your function, and T is what in this case? ...And..what class is this for?
     
  4. Oct 19, 2005 #3
    T mean Transposed, sorry I made a typo.


    Let L: R^2 ---> R^2 be a linear operator. If L((1,2)^T)) = (-2,3)^T
    and L((-1,1)) = (5,2)^T determine the value of L((7,5))^T

    This is some Linear Algebra homework I am stuck on.
     
  5. Oct 19, 2005 #4
    k. Well, I'll assume (-1,1) is transposed also.

    One way you can do this is by looking at the transformation matrix of L. Let's say its [x y]. Matrix multiplying ur vector by the transformation matrix should get you your answer. In this case, the values of x, y are not given but the answers are (by answers, i mean images). You should see that this becomes a problem of solving two equations.
     
  6. Oct 19, 2005 #5
    I still a little confused, please can you make it a little more clear?
     
  7. Oct 19, 2005 #6

    shmoe

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    Try writing (7,5) as a linear combination of (1,2) and (-1,1). How will this help?

    PS. it's ok to think of your vectors as row vectors, you could then leave out the transpose. Makes things a little neater in text.
     
  8. Oct 19, 2005 #7
    shmoe's idea is on the right track...sorry, but I was way off, i think.
     
  9. Oct 19, 2005 #8
    Try writing (7,5) as a linear combination of (1,2) and (-1,1). How will this help?

    I did, but I still don't see how it would help?
     
  10. Oct 19, 2005 #9
    well now that you've done that consider the definition of A linear transformation.

    A function L: R^n--->R^m is called a linear transformation or linear map if it satisfies

    i) L(u+v)= L(u) + L(v) for all u,v in R^n
    ii) L(cv)= cL(v) for all v in R^n, and scalar c

    Using both, this defintion and the combination you just made you should be able to get your answer.
     
  11. Oct 19, 2005 #10
    I just realized that the combination I created is one of a few different combinations, does that matter which combination I use? I still can't get an answer. Or rather, I still can't get the answer that matches the books. is my linear combination correct?
    where x1 = (1,1) and x2 = (2,-1)

    4*x1 + 3*x2 = 7
    3*x1 + (x2) = 5

    where x1 and x2 have been taken from the (1,2) and (-1,1)


    I might have figured something out,
    if I allow for a matrix multiplied by some other matrix, is that how i come about my answer?
     
    Last edited: Oct 19, 2005
  12. Oct 19, 2005 #11
    I really thank you guys for your help and patience with me.

    I think I figured out my answer, and it all makes sense. You guys are awsome.:rofl: :approve: :approve: :approve:
     
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