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Thread: [SOLVED] Linear substitution

  1. #1
    Senior Member Spec's Avatar
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    [SOLVED] Linear substitution

    D is the triangle with corners in (0,0), (1,-1) and (3,0).

    If we make the substitution: $\displaystyle u=x+y$ and $\displaystyle v=x-2y$ we end up with a triangle E with corners in (0,0), (0,3) and (3,3).

    The triangle D is formed by the three lines $\displaystyle x+y=0$, $\displaystyle x-2y=3$ and $\displaystyle y=0$, but I'm not quite sure how you're supposed to think when you transform this to uv coordinates (the E triangle).
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  2. #2
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    Quote Originally Posted by Spec View Post
    D is the triangle with corners in (0,0), (1,-1) and (3,0).

    If we make the substitution: $\displaystyle u=x+y$ and $\displaystyle v=x-2y$ we end up with a triangle E with corners in (0,0), (0,3) and (3,3).

    The triangle D is formed by the three lines $\displaystyle x+y=0$, $\displaystyle x-2y=3$ and $\displaystyle y=0$, but I'm not quite sure how you're supposed to think when you transform this to uv coordinates (the E triangle).
    just transform each sides of the triangle of D: the side $\displaystyle x+y=0, \ 0 \leq x \leq 1, \ -1 \leq y \leq 0$ goes to $\displaystyle u=x+y=0, \ v=x-2y=-3y.$ since $\displaystyle -1 \leq y \leq 0,$ we'll get: $\displaystyle 0 \leq v \leq 3.$

    so that side of D is transformed to $\displaystyle u=0, \ 0 \leq v \leq 3.$ now let's see what happens to the second side of D, i.e. $\displaystyle x-2y=3, \ 1 \leq x \leq 3, \ -1 \leq y \leq 0.$ we'll have: $\displaystyle v=x-2y=3$

    and $\displaystyle u=x+y=3y+3.$ since $\displaystyle -1 \leq y \leq 0,$ we'll have $\displaystyle 0 \leq 3y+3 \leq 3.$ so the second side of D goes to $\displaystyle 0 \leq u \leq 3, \ v = 3$ in uv plane. the third side of D is $\displaystyle y=0, \ 0 \leq x \leq 3.$ thus

    $\displaystyle u=x+y=x, \ v=x-2y=x.$ so $\displaystyle u=v=x.$ hence the image of the third side of D in uv plane is the line segment $\displaystyle v=u, \ 0 \leq u \leq 3.$
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