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!

Change of coordinate/Reflection Linear Algebra Problem

  1. Feb 21, 2007 #1
    Change of coordinate/Reflection Linear Algebra Problem!!

    1. The problem statement, all variables and given/known data
    Hi everyone, so here I am yet again :blushing:.

    In R^2 let L be the line y=mx, where m is not zero. Find an expression for T(x,y) where T is the reflection of R^2 about L.
    2. Relevant equations

    A transformation that reflects about the x axis is given by T(x,y) = (x, -y) while one about the y axis is T(x,y)=(-x,y).
    The change of coordinate matrix Q is given by x'_j = (summation) (Q_ij) (x_i)
    (sorry, not sure how to add symbols into these windows), and we probably also need the relationship that the matrix representation of [T]_B' = Q^-1*[T]_B*Q where B' and B are ordered bases for R^2.

    Also, the equation of a line in slope-intercept form is y = mx + b. A line orthogonal to this would be y = - x/m +b.

    3. The attempt at a solution

    I know that if we are talking about reflection about a line L with slope m, that reflection will occur for lines orthogonal to L with slope -1/m. I thought that I could set it up as follows:

    For any ordered pair (x,y) on L, let T(x,y) = (x,y) since the line itself is invariant under transformation. Then consider some (-x,y) on a line L' that is perpendicular to L (it could've been (x, -y), but it there is no loss of generality with what I have). For T(-x,y) = -(-x, y) = (x, -y).

    If we let B be the standard ordered basis for R^2, we can let B' = {(1,1),(-1,1)}. Then we get the change of coordinate matrix (noted here as columns) Q = ((1 1) (1 -1)}, and Q^-1 = {(-1 -1) (-1 1)}.

    For [T]_B we have ((1 0) (0,-1)}. Then it is easy to find [T]_B' = Q^-1*[T]_B*Q.

    From that answer, which should be a 2*2 matrix, it appears that T is left multiplication by [T]_B. Thus for any (x,y) in R^2 I can say

    T(x,y) = [T]_B*((x) (y)) = some set of equations.

    Basically, I am lost at the beginning point where I need to decide what B' and Q are. Once I get those, I think the rest of this will work. I kind of just guessed by saying that B'={(1,1),(-1,1)} -- I'm just not sure how to get this step to work.

    Or should my B' be something more like (x,y), (-x,y) with x and y in it? I am very confused about how to find a B'.

    For reference, the answer in the back of my books is:

    T(x,y) = (1/m^2)*((1-m^2)x + 2my, 2mx + (m^2 -1)y). I have NO idea how this ordered pair is supposed to emerge. :bugeye: Maybe my entire method is wrong. Can anyone help?
     
  2. jcsd
  3. Feb 23, 2007 #2
    Did you try rotating by an angle -\theta so that the line goes to an axis (say x-axis), followed by reflection, followed by rotating back by \theta? This amounts to a matrix product.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Change of coordinate/Reflection Linear Algebra Problem
Loading...