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Math Help - show that T has an inverse

  1. #1
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    show that T has an inverse

    let T: R^2 -> R^2 be a linear transformation define by


    T \left[ \begin{array}{c}x\\y\end{array} \right] =\left[ \begin{array}{c}x+y\\x-y \end{array} \right]


    they are matrix they are both 2x1 incase you were wondering wth i did xD

    show that T has an inverse

    alright so im pretty sure i have to show that the LS has an inverse for T to have an inverse but i never really did any any question with a 1x2 matrix so i dont know how to do the inverse.. can someone help me out please
    Last edited by CaptainBlack; December 12th 2009 at 01:27 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by treetheta View Post
    let T: R^2 -> R^2 be a linear transformation define by


    T \left[ \begin{array}{c}x\\y\end{array} \right] =\left[ \begin{array}{c}x+y\\x-y \end{array} \right]


    they are matrix they are both 2x1 incase you were wondering wth i did xD

    show that T has an inverse

    alright so im pretty sure i have to show that the LS has an inverse for T to have an inverse but i never really did any any question with a 1x2 matrix so i dont know how to do the inverse.. can someone help me out please
    Consider:

    T \left[ \begin{array}{c}x+y\\x-y\end{array} \right]

    CB
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  3. #3
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    i wish i knew where to go from there, but it's still not clear to me

    T is a linear trasformation that means it's composed of many elementary matrix's

    so E1E2 time the matrix  x1 = (x+y)^T and x2 = (x - y)^T
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  4. #4
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    Quote Originally Posted by treetheta View Post
    let T: R^2 -> R^2 be a linear transformation define by


    T \left[ \begin{array}{c}x\\y\end{array} \right] =\left[ \begin{array}{c}x+y\\x-y \end{array} \right]


    they are matrix they are both 2x1 incase you were wondering wth i did xD

    show that T has an inverse

    alright so im pretty sure i have to show that the LS has an inverse for T to have an inverse but i never really did any any question with a 1x2 matrix so i dont know how to do the inverse.. can someone help me out please
    Do you remember way back in "precalculus" where you learned to find the inverse of a function? To find the inverse of y= f(x), swap x and y to get x= f(y) and solve for y. That gives y= f^{-1}(x)
    You can do a similar thing here: Since T(x,y)= (x+y, x-y)= (a, b), swapping (x,y) and (a,b) gives you (a+b, a- b)= (x,y) or a+ b= x, a- b= y. Solve those two equations for a and b.
    Last edited by HallsofIvy; December 13th 2009 at 06:25 AM.
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  5. #5
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    but that is saying it has an inverse for sure right, i'm trying to prove that T has an inverse so does this qualify for a valid proof?
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    Senior Member Sampras's Avatar
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    Quote Originally Posted by treetheta View Post
    but that is saying it has an inverse for sure right, i'm trying to prove that T has an inverse so does this qualify for a valid proof?
    Show that it is 1-1 and onto.
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    Consider:

    T \left[ \begin{array}{c}x+y\\x-y\end{array} \right]

    CB
    Ok so you have not looked at this, lets make it more explicit:

    T \left[ T \left[ \begin{array}{c}x\\y\end{array} \right]\right] = T \left[ \begin{array}{c}x+y\\x-y\end{array} \right] =2\;\left[ \begin{array}{c}x\\y\end{array} \right]

    CB
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  8. #8
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    Quote Originally Posted by treetheta View Post
    but that is saying it has an inverse for sure right, i'm trying to prove that T has an inverse so does this qualify for a valid proof?
    Finding its inverse is definitely a valid way of showing that it has an inverse!
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