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Math Help - Problem on Linear Transformations

  1. #1
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    Problem on Linear Transformations

    Find all linear transformations T: R2 --> R2 which carry the line y=x to y=3x.

    Please help
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  2. #2
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    Re: Problem on Linear Transformations

    Hey arindamnaskr.

    Hint: Consider a rotation matrix taking a point on the line y = x to y = 3x by considering rotating both points by the difference of the angle between the two lines.
    Thanks from arindamnaskr
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  3. #3
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    Re: Problem on Linear Transformations

    Quote Originally Posted by chiro View Post
    Hey arindamnaskr.

    Hint: Consider a rotation matrix taking a point on the line y = x to y = 3x by considering rotating both points by the difference of the angle between the two lines.
    yes... I thought about that...but, the problem with that approach is finding the angle between those two lines...i mean, the angle is not something nice. So, I have been looking for an approach in which the matrix of T looks nice.

    Anyway, thank you for the reply. Please let me know if you come up with another approach.
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  4. #4
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    Re: Problem on Linear Transformations

    The angle between the two lines is \theta- \phi where tan(\theta)= 3 and tan(\phi)= 1, the slopes of the two lines. And
    tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}
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  5. #5
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    Re: Problem on Linear Transformations

    Quote Originally Posted by HallsofIvy View Post
    The angle between the two lines is \theta- \phi where tan(\theta)= 3 and tan(\phi)= 1, the slopes of the two lines. And
    tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}
    sometimes we miss the simplest calculations. I calculated the angle in a different way and it didn't look nice. Thanks, now it is done.
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  6. #6
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    Re: Problem on Linear Transformations

    Quote Originally Posted by arindamnaskr View Post
    Find all linear transformations T: R2 --> R2 which carry the line y=x to y=3x.
    Please help
    Take a point x,y on l1 and draw a line l through it with slope m. The intersection of l with l2 represents a linear transformation of l1 to l2 for arbitrary m (all linear transformations from l1 to l2). Then solve

    T(x,y) = (x,y)
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