1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Another Linear Transformation

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

    This is a slight variation of the last problem I posted.

    Write the standard matrix representation for T1T2 and use it to find [T1T2(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]

    [tex]
    T_2\left(x_1,x_2,x_3\right)=\left(x_3-x_1,x_3-2x_2-x_1,x_1-x_3\right)
    [/tex]

    3. The attempt at a solution

    [tex]
    T_1T_2=\left(x_3,-x_1,x_3\right)\cdot \left(x_3-x_1,x_3-2x_2-x_1,x_1-x_3\right)=x_1^2+2 x_1 x_2-x_1 x_3
    [/tex]

    [tex]
    A=\left(x_1^2+2 x_1 x_2-x_1 x_3\right)\left(
    \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1
    \end{array}
    \right)
    [/tex]

    Will A just end up being an identity matrix multiplied by the scalar that results from T1T2, or should I use a non-standard product for T1T2?
     
    Last edited: Jul 19, 2009
  2. jcsd
  3. Jul 19, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You have it all wrong. T1*T2(x) means T1(T2(x)). That's nothing like T1(x)*T2(x) whatever that means. Find the matrices corresponding to T1 and T2 and multiply them.
     
  4. Jul 19, 2009 #3
    Is this right, then?

    [tex]
    A=\left(
    \begin{array}{ccc}
    0 & 0 & 1 \\
    -1 & 0 & 0 \\
    0 & 0 & 1
    \end{array}
    \right)\cdot \left(
    \begin{array}{ccc}
    -1 & 0 & 1 \\
    -1 & -2 & 1 \\
    1 & 0 & -1
    \end{array}
    \right)=\left(
    \begin{array}{ccc}
    1 & 0 & -1 \\
    1 & 0 & -1 \\
    1 & 0 & -1
    \end{array}
    \right)
    [/tex]
     
  5. Jul 19, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Almost. But why are there two 1's in the third column of the first matrix?
     
  6. Jul 19, 2009 #5
    Thanks!

    There are two 1's in the third column of the first matrix because there is an x3 in the first and last element of T1 (is that the right terminology?)
     
  7. Jul 19, 2009 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You were right. My mistake.
     
  8. Jul 19, 2009 #7

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I made a mistake with that same transformation on another thread. There's something about that transformation... it's pretty sneaky
     
  9. Jul 19, 2009 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Must be us. DanielFaraday didn't have a problem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook