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Math Help - Linear transformations

  1. #1
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    Linear transformations

    1) Verify that the following transformations are linear

    a) T:R^2=>M(2,2)
    T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y \end{pmatrix}
    How can I solve?

    b) T:R^2
    T(x,y)=(x^2+y^2,x)
    Not is Linear transformations:
    f(u+v)=((x1)^2+2x1x2+(x2)^2+(y1)^2+2y1y2+(y2)^2,x1  +x2)

    c) T:R^3=>R^3
    T(x,y,z)=(x+y,x-y,0)
    Yes this is Linear transformation

    d) T:R=>R^2
    T(x)=(x,2)
    Yes this is Linear transformation


    How to handle first? The rest this correct?
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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    1) Verify that the following transformations are linear

    a) T:R^2=>M(2,2)
    T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y \end{pmatrix}
    How can I solve?
    Remember what linear means.
    You need to show T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y}).

    Now if \bold{x} = (x_1,x_2) and \bold{y} = (y_1,y_2) does T((x_1+y_1,x_2+y_2)) = T(x_1,x_2)+ T(y_1,y_2).

    And you also need to show T(k\bold{x}) = kT(\bold{x}) in a similar way.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Remember what linear means.
    You need to show T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y}).

    Now if \bold{x} = (x_1,x_2) and \bold{y} = (y_1,y_2) does T((x_1+y_1,x_2+y_2)) = T(x_1,x_2)+ T(y_1,y_2).

    And you also need to show T(k\bold{x}) = kT(\bold{x}) in a similar way.

    I know this was what I did. And about my doubts?
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  4. #4
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    Quote Originally Posted by Apprentice123 View Post
    I know this was what I did. And about my doubts?
    T(x_1,x_2) = \begin{bmatrix}2x_2&3x_1\\-x_2&x_1+2x_2\end{bmatrix}

    T(y_1,y_2) = \begin{bmatrix}2y_2&3y_1\\-y_2&y_1+2y_2\end{bmatrix}

    T(x_1,x_2)+T(y_1,y_2) = \begin{bmatrix} 2(x_2+y_2) & 3(x_1+y_1) \\ - (x_2+y_2) & (x_1+y_1) + 2 (x_2+y_2) \end{bmatrix} = T(x_1+y_1,x_2+y_2).

    Thus, the first condition is true.
    Proving the second one is a lot easier.
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