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Math Help - Linear transformation question

  1. #1
    Junior Member
    Joined
    Nov 2009
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    38

    Linear transformation question

    Plz help me for answer for the following question;

    T is a linear transformation of \mathbb{R} ^3 into \mathbb{R} ^3, such that

    T \begin{pmatrix}<br />
  1      \\<br />
  1  \\ <br />
  1<br />
\end{pmatrix} =<br />
\begin{pmatrix}<br />
  3      \\<br />
  4 \\ <br />
  4<br />
\end{pmatrix} , <br /> <br />

    T \begin{pmatrix}<br />
  1      \\<br />
  2 \\ <br />
  3<br />
\end{pmatrix} = <br />
\begin{pmatrix}<br />
  6      \\<br />
  7 \\ <br />
  8<br />
\end{pmatrix} , <br /> <br />

    T \begin{pmatrix}<br />
  1      \\<br />
  0 \\ <br />
  1<br />
\end{pmatrix} = <br />
\begin{pmatrix}<br />
  2      \\<br />
  3 \\ <br />
  2<br />
\end{pmatrix}<br /> <br />

    Find T and Kernal of T

    When answering the question, I found taking linear transformation
    T(x_1 , x_2 ,x_3)= x_1 T(1,1,1)+ x_2 T(0,1,0)+ x_3 T(0,0,1)<br />
     = x_1(3,4,4)+ x_2 (6,7,8)+ x_3 (2,3,2)<br />
=(3 x_1 +6 x_2+2 x_3 , 4x_1+8x_2+3x_3, 4x_1+7x_2+2x_3)

    After this how I going to calculate T and kernal of T
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  2. #2
    Banned
    Joined
    Oct 2009
    Posts
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    Thanks
    2
    Quote Originally Posted by dhammikai View Post
    Plz help me for answer for the following question;

    T is a linear transformation of \mathbb{R} ^3 into \mathbb{R} ^3, such that

    T \begin{pmatrix}<br />
1 \\<br />
1 \\ <br />
1<br />
\end{pmatrix} =<br />
\begin{pmatrix}<br />
3 \\<br />
4 \\ <br />
4<br />
\end{pmatrix} , <br /> <br />

    T \begin{pmatrix}<br />
1 \\<br />
2 \\ <br />
3<br />
\end{pmatrix} = <br />
\begin{pmatrix}<br />
6 \\<br />
7 \\ <br />
8<br />
\end{pmatrix} , <br /> <br />

    T \begin{pmatrix}<br />
1 \\<br />
0 \\ <br />
1<br />
\end{pmatrix} = <br />
\begin{pmatrix}<br />
2 \\<br />
3 \\ <br />
2<br />
\end{pmatrix}<br /> <br />

    Find T and Kernal of T

    When answering the question, I found taking linear transformation
    T(x_1 , x_2 ,x_3)= x_1 T(1,1,1)+ x_2 T(0,1,0)+ x_3 T(0,0,1)<br />
     = x_1(3,4,4)+ x_2 (6,7,8)+ x_3 (2,3,2)<br />
=(3 x_1 +6 x_2+2 x_3 , 4x_1+8x_2+3x_3, 4x_1+7x_2+2x_3)

    After this how I going to calculate T and kernal of T

    Well, you did most of the hardest part: now you can take the matrix of T wrt the standard basis: \begin{pmatrix}3&6&2\\4&8&3\\4&7&2\end{pmatrix} and bring it to echelon form: that'll will give you a solution to the associated homogeneous linear system, i.e. the kernel of the matrix (of T), and then you can take a basis for the row space for the image of T (all the time, wrt the standard basis) .

    Tonio
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  3. #3
    Junior Member
    Joined
    Nov 2009
    Posts
    38
    Hi thanks for the help, Now I got an idea for the solution, thanks again..
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