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Linear Transformations and Matrices

  1. Mar 6, 2009 #1
    I'm having some difficulty understanding how to perform linear transformations on matrices. I understand the definition but not how to perform the operations. I'm going to give a few examples from my book:

    Suppose that [tex] T: R^2 \longrightarrow R^2[/tex] is a linear transformation such that.

    [tex]T\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right)=\left( \left[\begin{array}{cc}2\\3\end{array}\right]\right)[/tex]

    And

    [tex]T\left(\left[\begin{array}{cc}1\\-1\end{array}\right]\right)= \left(\left[\begin{array}{cc}4\\-1\end{array}\right]\right)[/tex]


    (a) Find: [tex] T\left(\left[\begin{array}{cc}3\\3\end{array}\right]\right)[/tex]


    Solution:


    since:
    [tex]\left(\left[\begin{array}{cc}3\\3\end{array}\right]\right) = 3\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right)[/tex]

    it follows that

    [tex]T\left(\left[\begin{array}{cc}3\\3\end{array}\right]\right) = T3\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right) = 3T\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right) = 3\left(\left[\begin{array}{cc}2\\3\end{array}\right]\right) = \left(\left[\begin{array}{cc}6\\9\end{array}\right]\right)[/tex]


    How do they come to this solution?
     
  2. jcsd
  3. Mar 6, 2009 #2
    it follows from the definition of a linear transformation:

    If T is a linear transformation, then
    [tex]T(av_1 +bv_2) = aT(v_1)+bT(v_2)[/tex]

    a and b are constants and [itex]v_1[/itex] and [itex]v_2[/itex] are vectors.
     
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