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Math Help - Linear transformations

  1. #1
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    Linear transformations

    check that it is linear Transformation

    T:R^2 =>M(2,2)

    T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix}
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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    check that it is linear Transformation

    T:R^2 =>M(2,2)

    T(x,y)=\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix}
    if T is a linear transformation then it must be closed under addition (T1) and scalar multiplication (T2).

    For T1. T(x,y + x_1,y_1) = \begin{pmatrix} 4y & 6x \\ -2y & 2x+4y\end{pmatrix} =

     \begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix} + \begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix} = T(x,y) + T(x_1, x_2)

    For T2. T(kv) = \begin{pmatrix} 2yk & 3xk \\ -yk & k(x+2y)\end{pmatrix} = k\begin{pmatrix} 2y & 3x \\ -y & x+2y\end{pmatrix}= kT(v)

    Therefore the transformation T is a linear transformation.
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