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Math Help - finding the 2 x 2 matrix

  1. #1
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    finding the 2 x 2 matrix

    If I get a question in the exam that shows any transformation, and then asks me to find the "2 x 2 matrix which represents the transformation", how would I find it? Is this a case of memorizing specific matrices?

    Here's a real question from a past paper:




    (a) Describe fully the single transformation which maps triangle A onto triangle B.

    Rotation, y = x

    (b) Find the 2 2 matrix which represents this transformation.
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  2. #2
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    It's actually REFLECTION about the line y = x.

    What transformation matrix represents reflection?
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    That's the thing, I don't know.
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  4. #4
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    You know that this particular reflection swaps the x and y coordinates, so:


    you know from matrix multiplication that this means:
    ax1 + bx2 = x2
    => a=0, b=1


    cx1+dx2 = x1
    =>c=1,d=0

    so the transformation matrix is:
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  5. #5
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    Quote Originally Posted by SpringFan25 View Post
    You know that this particular reflection swaps the x and y coordinates, so:
    How do I know this?

    you know from matrix multiplication that this means:
    ax1 + bx2 = x2
    => a=0, b=1
    I don't understand how you did this sum. How does ax1 + bx2 = x2 lead me to know the values of a and b?
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  6. #6
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    A reflection about the line y = x is the same as finding an inverse function, in which the domain and range are swapped - in other words, the x and y values are swapped.

    Surely if ax_1 + b_2 = x_2, the right hand side is 0x_1 + 1x_2...
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  7. #7
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    Quote Originally Posted by Prove It View Post
    A reflection about the line y = x is the same as finding an inverse function, in which the domain and range are swapped - in other words, the x and y values are swapped.
    Okay thanks I get that now. :-)

    Surely if ax_1 + b_2 = x_2, the right hand side is 0x_1 + 1x_2...
    Sorry I must be the stupidest person alive, but I don't know what you mean by this! Could you explain it fuller? What right hand side are you talking about?
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  8. #8
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    The right hand side of ax_1 + bx_2 = x_2...

    Surely x_2 is the same as 0x_1 + 1x_2...

    Therefore ax_1 + bx_2 = 0x_1 + 1x_2

    What are a and b?
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  9. #9
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    Oh ok, a = 0 and b = 1. So you basically just assigned two random values to a and b?

    Sorry Im asking so many questions. I understand the basics of matrices (addition, multiplication, determinants, inverse) but transformational matrices seem to go over my head, and my textbook is terrible at explaining it.
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  10. #10
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    No we did not assign two random values to a and b. For the equation to be balanced, the x_1 terms have to have the same coefficients, and the x_2 terms have to have the same coefficients.

    So if ax_1 + bx_2 = x_2, there is clearly a zero x_1 term. So a = 0 and b = 1.
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    Ok thanks that makes perfect sense now! I think I've got it now. Thanks for your help. :-)
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    Is there a sort of guide online that you can point me to that gives a point by point instruction on transformations?
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  13. #13
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    Wikipedia is a good place to start.
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  14. #14
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    Ok perhaps I didn't understand it so well. Here's another sum I've got:

    ( 4.. is transformed into ( 8
    ..3 ) ............................6 )

    So my workings are...
    4a + 3b = 8

    So now I'm stuck and I know I've missed something. Help, please!
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  15. #15
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    What is the transformation doing? It's enlarging both the x and y values by a factor of 2. So clearly matrix is ..... Think about how the old x values are related to the new ones and the same with the y values.
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