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Math Help - Linear Transformations

  1. #1
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    Linear Transformations

    Let u = (1,2), v = (3,1) and T: R^{2}\rightarrow R be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)


    I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?
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  2. #2
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    <br />
S=[\vec{u},\vec{v}],\begin{bmatrix}-3\\4\end{bmatrix}=S(S^{-1}\begin{bmatrix}-3\\4\end{bmatrix})=[\vec{u},\vec{v}]\begin{bmatrix}3\\-2\end{bmatrix}= 3\vec{u} -2\vec{v}

    <br />
T(\begin{bmatrix}-3\\4\end{bmatrix})=T(3\vec{u} -2\vec{v})=3T(\vec{u})-2T(\vec{v})=3\times 4-2\times 5=2<br />
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  3. #3
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    Quote Originally Posted by temaire View Post
    Let u = (1,2), v = (3,1) and T: R^{2}\rightarrow R be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.)


    I don't really know where to start. I know I have to write (-3,4) as a linear combination of u and v, but what do I do from there?
    Once you have (-3, 4)= au+ bv, T((-3,4))= aT(u)+ bT(v)= 4a+ 5b.
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