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Math Help - Linear transformation

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    35

    Linear transformation

    Consider the basis S={ v1,v2 } where v1= (-2,1), v2=(1,3) and let
    T : R^2 --> R^3 be the linear transformation such that

    T(v1) = ( -1,2,0) and T(v2) = (0,-3,5)

    Find a formula for T( x1, x2)

    Here what I did

    let x = (x1,x2) and

    c1v1+c2v2=x so

    -2c1 +1c2 = x1
    1c1 + 3c2 = x2 so i have to express c1 and c2 in term of x1 and x2

    i get this system

    -2 1 = x1
    1 3 = x2
    I use ref(a) to get this

    1 -1/2 = x1
    0 1 = x2
    so c2= x2
    c1= x1+1/2x2

    (x1 +1/2x2)(-1,2,0) + (x2)(0,-3,5) = (-x1-1/2x2, 2x1-2x2, 5x2)

    the answer is 1/7(3x1-x2, -9x1-4x2, 5x1+10x2)

    what did i do wrong here? thanks in advanced
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  2. #2
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303
    Hey there,

    You have the right idea, you just messed up on your algebra. Try redoing your first system of equations.

    Working down to RREF I got...

    \begin{bmatrix}1&0\\0&1\end{bmatrix} <br />
=\begin{bmatrix}\frac{x_{2}}{7}-\frac{3x_{1}}{7}\\\frac{2x_{2}}{7}+\frac{x_{1}}{7}  \end{bmatrix}

    T(\vec{x_{1}},\vec{x_{2}})=c_{1}(-1,2,0)+c_{2}(0,-3,5)= your books answer!
    Last edited by Danneedshelp; July 27th 2009 at 07:35 PM.
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