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RadiationX
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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?
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?