1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Linear Maps

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Determine the matrix A for the linear map T: R3R3 which is defined by that the vector u first is mapped on v×u, where v=(-9,2,9) and then reflected in the plane x=z (positively oriented ON-system). Also determine the determinant for A.

    2. Relevant equations

    3. The attempt at a solution
    I actually started with the determinant and said that since the first mapping is a projection the determinant of that is =0 -- thus, the determinant for the whole thing is 0, since det(B*C)=det(B)*det(C) and in this case, A=S*P, where S is the determinant for the reflection (which is what we usually use in Swedish) and P is the projection.

    Anyway, I started with S (for practise, if nothing else):

    The plane will have the equation x-z=0 in its normal form and thus the normal to the plane is <1,0,-1>. So if I call vector w <a,b,c>, the reflection in the plane is:

    <a,b,c> + t<1,0,-1> = <a+t,b,c-t>.

    <a,b,c> + (t/2)<1,0,-1> needs to be on the plane and thus the coordinates for that vector need to fulfill the plane equation,

    (a+t/2) - (c-t/2) = 0, t= -(a-c).


    Thus, the reflection matrix is:
    [tex]S = \left( \begin{array}{ccc}
    0 & 0 & 1 \\
    0 & 1 & 0 \\
    1 & 0 & 0 \end{array} \right)\]

    I didn't think that was too difficult, but now I'm entering the confusing part.

    I can quite easily calculate v×u, if u=<x,y,z>.

    (v×u = < -9y+2z, 9x+9z, -2x - 9y>.)

    But (and I was thinking this from the very beginning) v×u is orthogonal to u (by definition of the cross product, because it creats a vector orthogonal to the plane containing v and u). But there can't be any projection if it's orthogonal, right? So I thought that if they all stay the same it should be the identity matrix and thus S*I=S, but that is not correct. And now I've got no idea what to do... And I didn't really want it to be the identity matrix, because that would not mean that the determinant is 0.

    The conclusion is that I'm very confused.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted