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Math Help - Linear Transformation Question

  1. #1
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    Linear Transformation Question

    Let v be a fixed vector in R^3. Show that the transformation defined by T(u) = v x u is a linear transformation.


    I am confused because is this a special case where V = W(by definition), so the linear transformation is a linear operator?

    If that is the case how do i show this is a linear transformation?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by flaming View Post
    Let v be a fixed vector in R^3. Show that the transformation defined by T(u) = v x u is a linear transformation.


    I am confused because is this a special case where V = W(by definition), so the linear transformation is a linear operator?

    If that is the case how do i show this is a linear transformation?
    You can try to show that this is a linear transformation in one step by checking if T\left(\alpha u_1+\beta u_2\right)=\alpha T\left(u_1\right)+\beta T\left(u_2\right), where u_1,u_2\in\mathbb{R}^3:

    T\left(\alpha u_1+\beta u_2\right)=\left(\alpha u_1+\beta u_2\right)\times v

    Now from cross product properties:

    \left(\mathbf{u}+\mathbf{v}\right)\times\mathbf{w}  =\mathbf{u}\times\mathbf{w}+\mathbf{v}\times\mathb  f{w} and \left(k\mathbf{u}\right)\times\mathbf{v}=k\left(\m  athbf{u}\times\mathbf{v}\right)

    Thus,

    \begin{aligned}T\left(\alpha u_1+\beta u_2\right)&=\left(\alpha u_1+\beta u_2\right)\times v\\&=\left(\alpha u_1\right)\times v+\left(\beta u_2\right)\times v\\&=\alpha\left(u_1\times v\right)+\beta \left(u_2\times v\right)\\&=\alpha T\left(u_1\right)+\beta T\left(u_2\right)\end{aligned}

    Thus, T is a linear transformation.

    Does this make sense?
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