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Check work on linear transformation problem

  1. Oct 16, 2005 #1
    check work plz on linear transformation problem

    The problem is to find a standard matrix of T.
    [tex]T:\mathbb{R}^3\rightarrow\mathbb{R}^2, T(\vec{e}_1) = (1,3), T(\vec{e}_2) = (4,-7), T(\vec{e}_3) = (-5,4)[/tex]
    where e_1, e_2, and e_3 are the columns of the 3x3 identity matrix.
    So here's what I did:
    Find A for [itex]T(\vec{x})=A\vec{x}[/itex]

    [tex]\vec{x}=I_3\vec{x}=\left[ \vec{e}_1 \; \vec{e}_2 \; \vec{e}_3 \right] \vec{x} = x_1\vec{e}_1 + x_2\vec{e}_2 + x_3\vec{e}_3[/tex]

    [tex]T(\vec{x}) = T(x_1\vec{e}_1 + x_2\vec{e}_2 + x_3\vec{e}_3) = x_1 T(\vec{e}_1) + x_2 T(\vec{e}_2) + x_3 T(\vec{e}_3)[/tex]

    [tex]\left[ T(\vec{e}_1) \; T(\vec{e}_2) \; T(\vec{e}_3)\right]\vec{x} = A\vec{x} [/tex]

    [tex] A = \left[ T(\vec{e}_1) \; T(\vec{e}_2) \; T(\vec{e}_3)\right] = [/tex]
    (well basically i screwed up teh tex for this but its a 2x3 matrix with top row 1,4,-5 and bottom row 3,-7,4)
    Note: a lot of this might be unnecessary, but my main goal is that I want to be sure that I am understanding this correctly.
    Last edited: Oct 16, 2005
  2. jcsd
  3. Oct 16, 2005 #2
    I think I finally understand linear transformation. Can someone check this one also?

    [itex]T:\mathbb{R}^2\rightarrow\mathbb{R}^2[/itex] first reflects points through the horizontal [itex]x_1[/itex]-axis and then reflects points over the line [itex] x_2=x_1[/itex]

    what I did:

    Find A for [itex]T(\vec{x})=A\vec{x}[/itex]

    Let [itex]T_1[/itex] be the first reflection.
    [tex]\vec{e}_1 = (1,0), \vec{e}_2 = (0,1)[/tex]
    [tex]T_1(\vec{e}_1) = (1,0), T_1(\vec{e}_2) = (0,-1)[/tex]

    Let [itex]T_2[/itex] be the second reflection.
    [tex]T(\vec{e}_1) = T_2(T_1(\vec{e}_1)) = (0,1), T(\vec{e}_2) = T_2(T_1(\vec{e}_2)) = (-1,0) [/tex]
    [tex]A = [T(\vec{e}_1) \; T(\vec{e}_2)] = \left[
    0 & 1\\
    -1 & 0
    \right] [/tex]
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