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Find a 2x2 Matrix which performs the operation...

  1. Jun 20, 2015 #1
    1. The problem statement, all variables and given/known data




    Find the matrix that performs the operation

    2x2 Matrix which sends e1→e2 and e2→e1


    2. Relevant equations


    3. The attempt at a solution

    I know e1 = < 1 , 0>
    and e2 = <0 , 1>

    Basically I'm not quite sure what the question is asking. This is the one of the problems im currently stuck on. He is teaching out of the book-currently

    Can someone please explain what this means?
     
  2. jcsd
  3. Jun 20, 2015 #2

    Orodruin

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    The problem asks you to find a matrix ##A## such that ##A e_1 = e_2## and ##Ae_2 = e_1##.
     
  4. Jun 20, 2015 #3
    thanks. So If i understand from your clarification,

    This 2x2 matrix that I'm looking for, when I multiply it by e1 it will give me e2, and vice-versa, this unknown matrix multipled by e2 will give me the result e1.
     
  5. Jun 20, 2015 #4

    Orodruin

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  6. Jun 20, 2015 #5
    Thanks again.

    When i find the 2 these two matrixes that perform these two operations. When I multiply them together the result should be the 2x2 matrix that this question is asking for?
     
  7. Jun 20, 2015 #6

    Orodruin

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    No. You need to find a single matrix which satisfies both relations.
     
  8. Jun 20, 2015 #7
    How is that even possible?
    1] I found a 2x2 matrix we'll call A that when you multiply it by e1 = e2
    2] I also found a 2x2 matrix B that when you multiply it by e2 =e1

    But how is it possible to satisfy both conditions at the same time with the same 2x2 matrix? Any hints?
     
  9. Jun 20, 2015 #8

    Orodruin

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    Any matrix is defined by its action on a complete basis. If you just know how it acts on a single vector you cannot determine it uniquely. I suggest assuming the most general form
    $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$
    and examine what you can say about ##a,b,c## and ##d## from your requirements.
     
  10. Jun 20, 2015 #9
    Nevermind I just figured it out. thanks for your help with adding to my knowledge/intuition with that statement about defining a matrix actions.
    !(took me long enough)

    The 2x2 matrix i came up with was the following.
    num2_zpsyknllmyo.jpg
     
  11. Jun 23, 2015 #10

    WWGD

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    Like Orodruin was saying, a matrix associated to a linear map T is uniquely defined by the effects of T on the given choice of basis. Then , for a map (Assummiing this, given your posts) from ##\mathbb R^2 ## to itself , both with the same basis, the associated matrix is ## [(Te_1)^T , (Te_2)^T] ##. where the T means transpose. Note that the copies of ##\mathbb R^2 ## may have different bases, then the solution is different. There are many variants, of course, of maps between vector spaces ( or even --free, of course-- rings/modules) of different dimensions, with different bases, but the idea of your case generalizes nicely here.
     
    Last edited: Jun 23, 2015
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