Results 1 to 2 of 2

Math Help - Linear Transformation Question

  1. #1
    Banned
    Joined
    Jan 2009
    Posts
    20

    Linear Transformation Question

    Let is the linear transformation corresponding to a rotation by an angle of about the x -axis in and let Let is the linear transformation corresponding to a rotation by an angle of about the z -axis

    For what values of and does
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi

    Rotations are invertible, their inverse being the rotation by the opposite angle about the same axis.
    So T_2.T_1=T_1.T_2\ \Leftrightarrow\ T_1^{-1}.T_2.T_1=T_2

    Observe the action on x of both maps. \mathbb{R}x is left invariant by T_1 and T_1^{-1} and is the only subset which has that property.

    Thus a necessary condition to the equality is T_1^{-1}(T_2((x))=T_2(x)\ \text{i.e.}\ T_2(x)\in \mathbb{R}x, which means \vartheta_2=0 or \vartheta_2=\pi.

    A similar proof gives: \vartheta_1=0 or \vartheta_1=\pi.


    Therefore a necessary condition is: ((\vartheta_1=0\ \text{or}\ \vartheta_1=\pi)\ \text{and}\ (\vartheta_2=0\ \text{or}\ \vartheta_2=\pi))

    The only thing to show now is that in theese cases, T_1 and T_2 commute; so the condition will also be sufficient and thus equivalent to the equality.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. linear transformation question
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 19th 2011, 04:10 PM
  2. Linear transformation question
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 23rd 2010, 10:57 AM
  3. Linear transformation question
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 22nd 2009, 09:04 PM
  4. Linear Transformation Question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 6th 2008, 11:23 AM
  5. Linear Transformation Question.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 23rd 2007, 12:22 AM

Search Tags


/mathhelpforum @mathhelpforum