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Find a matrix that does this

  1. Feb 3, 2008 #1
    1. The problem statement, all variables and given/known data

    find a single matrix that performs the indicated successions of operations:
    expands by a factor of 5 in the y-direction, then shears with factor 2
    in the y-direction

    2. Relevant equations



    3. The attempt at a solution

    first for the expansion:
    (x,y) maps to (x,5y)
    then for the shear:
    (x,5y) maps to (x, (2x+5y)

    i think its right but im not quite sure.
    thank you.
     
  2. jcsd
  3. Feb 3, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You have written down the transformations correctly but you haven't answered the question! You were asked to find a matrix. What matrix does that?

    What matrix changes (x,y) to (x, 2x+ 5y)? In other words, find the a, b, c,d such that
    [tex]\left[\begin{array}{cc} a & b \\ c & d\end{array}\right]\left[\begin{array}{c} x \\ y\end{array}\right]= \left[\begin{array}{c} x \\ 2x+ 5y\end{array}\right][/tex]
    Multiplying the left side will give you two equations for a, b, c, d but remember they must be true for all x and y. Comparing corresponding coefficients will give you four very simple equations for the a, b, c, d.
     
    Last edited: Feb 3, 2008
  4. Feb 3, 2008 #3
    ok so multipying the left side gives the matrix:
    [(ax+by),(cx+dy)]
    so ax+by=x
    cx+dy=2x+5y

    now what? ;)
     
  5. Feb 3, 2008 #4
    You might separate the equations, in order to get:
    ax=x, by=0, cx=2x and dy=5y, which ought to be solvable.
     
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