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Homework Help: Linear Transformation matrix help

  1. Feb 16, 2010 #1
    The problem is as follows:
    Find a nonzero 2x2 matrix A such that Ax is parallel to the vector
    [1]
    [2]
    for all x in R2.

    So far I know A=[v1 v2] therefore Ax= [v1 v2][x1]
    [x2]

    = x1v1+x2v2
    I know these two vectors are parallel, but I am a little stuck how to relate this property to solve for the matrix A.
     
  2. jcsd
  3. Feb 17, 2010 #2

    Mark44

    Staff: Mentor

    You have Ax = k[1 2]T for any vector x in R2.
    Specify values for A.

    What does A do to [1 0]T? to [0 1]T?
     
  4. Feb 17, 2010 #3

    HallsofIvy

    User Avatar
    Science Advisor

    What Mark44 suggests works fine. A more "primitive method" is to write A as
    [tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]
    so your equation "Ax= k[1 2]T" becomes
    [tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]\begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix}ax+ by \\ cx+ dy\end{bmatrix}= \begin{matrix}k \\ 2k\end{bmatrix}[/tex]

    giving you two equations, ax+ by= k and cx+ dy= 2k for the 5 unknown numbers. There will, of course, be an infinite number of possible answers. You are simply asked to find one such matrix.
     
  5. Feb 17, 2010 #4

    Mark44

    Staff: Mentor

    Note: fixed your LaTeX by adding a missing [ tex] tag.
    That's what I meant by saying "specify values for A." I think we're on the same page here.
     
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