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Linear Transformation

  1. Jul 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Write the standard matrix representation for T1 and use it to find [T1(1,-3,0)]E.

    2. Relevant equations

    [tex]
    T_1\left(x_1,x_2,x_3\right)=\left(x_3,-x_1,x_3\right)
    [/tex]

    3. The attempt at a solution

    I just wanted to check to see if I am doing this right. Thanks in advance!

    [tex]
    A=\left(
    \begin{array}{ccc}
    0 & 0 & 1 \\
    -1 & 0 & 0 \\
    0 & 0 & 1
    \end{array}
    \right)\
    [/tex]

    [tex]
    \left[T_1(1,-3,0)\right]_E=A\left(
    \begin{array}{c}
    1 \\
    -3 \\
    0
    \end{array}
    \right)=\left(
    \begin{array}{ccc}
    0 & 0 & 1 \\
    -1 & 0 & 0 \\
    0 & 0 & 1
    \end{array}
    \right).\left(
    \begin{array}{c}
    1 \\
    -3 \\
    0
    \end{array}
    \right)=\left(
    \begin{array}{c}
    0 \\
    -1 \\
    0
    \end{array}
    \right)
    [/tex]
     
  2. jcsd
  3. Jul 19, 2009 #2

    Office_Shredder

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    Staff Emeritus
    Science Advisor
    Gold Member

    Your A is transposed from what it should be.
     
  4. Jul 19, 2009 #3

    HallsofIvy

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    Science Advisor

    No, Officeshredder,
    Applying T1 to each basis vector in turn gives the columns.

    T1(1, 0, 0)= (0, -1, 0)
    T1(0, 1, 0)= (0, 0, 0)
    T1(0, 0, 1)= (1, 0, 1)

    So T1 is represented by
    [tex]\begin{bmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 1 & 0 & 1 \end{bmatrix}[/tex]
    exactly what DanielFaraday has.

    And, of course, T1(1,-3,0)= (0,-1,0) as said.
     
  5. Jul 19, 2009 #4
    Thank you both for your input!
     
  6. Jul 19, 2009 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Office shredder may be using a different convention than you and I:

    [tex]T_1(1,-3,0)= \begin{bmatrix}1 & -3 & 0\end{bmatrix}\begin{bmatrix}1 & -1 & 0\\ 0 & 0 & 0 \\ 1 & 0 & 1\end{bmatrix}= \begin{bmatrix}0 \\ -1 \\ 0\end{bmatrix}[/tex]
     
  7. Jul 19, 2009 #6
    Yes, these things often depend on the textbook. Thanks.
     
  8. Jul 19, 2009 #7

    Office_Shredder

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    Staff Emeritus
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    Gold Member

    No, sorry, that was just a brain fart on my part
     
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