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Math Help - Linear Transformation

  1. #1
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    Linear Transformation

    Prove whether or not these functions are linear transformation.

    a) F:V_3(R) \to V_2(R), defined by F(a_1, a_2, a_3)=(a_1+a_2, a_1a_3-a_2)

    b) G:V_3(R) \to V_2(R), defined by G(a_1, a_2, a_3)=(a_1-a_2, 2a_1-a_3)
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  2. #2
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    A transformation, f, is linear if and only if it satisfies f(au+ bv)= af(u)+ bf(v) for any vectors u and v and any numbers a and b.

    If F(a_1, a_2, a_3)= (a_1+ a_2, a_1a_3- a_2) then what is F(a_1+ b_1, a_2+ b_2, a_3+ b_3). Is it the same as F(a_1,a_2,a_3)+ F(b_1, b_2, b_3)?
    Last edited by mr fantastic; May 4th 2009 at 03:17 AM. Reason: Fixed a subscript
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  3. #3
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    Yeah I don't know how to show whether it is Linear or not.
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  4. #4
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    Quote Originally Posted by LL_5 View Post
    Yeah I don't know how to show whether it is Linear or not.
    You have been told what to do. Try answering the question you were asked in post #2. Where do you get stuck?
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  5. #5
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    This is what I have done so far to prove whther it is linear:
    Attached Thumbnails Attached Thumbnails Linear Transformation-new.bmp  
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  6. #6
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    <br /> <br />
F(a_1, a_2, a_3)=(a_1+a_2, a_1a_3-a_2)<br />

    Let v_1= (x_1,y_1,z_1) and v_2=(x_2,y_2,z_2)

    A map is linear if F(\alpha_1 v_1+\alpha_2 v_2)=\alpha_1F(v_1)+\alpha_2F(v_2)

    F \left( \alpha_1 (x_1,y_1,z_1)+\alpha_2(x_2,y_2,z_2) \right)=F(\alpha_1 x_1+ \alpha_2 x_2, \alpha_1y_1+\alpha_2y_2, \alpha_1 z_1+ \alpha_2 z_2) =(\alpha_1 x_1+\alpha_2 x_2 +\alpha_1 y_1+\alpha_2 y_2, \alpha_1^2 x_1 z_1 + \alpha_2 \alpha_1 x_2 z_1+ \alpha_1 \alpha_2 x_1 z_2 + \alpha_2^2 x_2 z_2-\alpha_1y_1-\alpha_2y_2)

    \alpha_1F(v_1)+\alpha_2F(v_2)=\alpha_1(x_1+y_1, x_1 z_1-y_1)+ \alpha_2 (x_2+y_2, x_2 z_2-y_2)

    But F(\alpha_1 v_1+\alpha_2 v_2) \neq \alpha_1F(v_1)+\alpha_2F(v_2) so F is not a linear transformation.

    Now you try and do G. I think it is linear.....
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