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Math Help - Linear map question

  1. #1
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    Linear map question

    The question:

    Let b be a fixed vector in \mathbb{R}^3. Is the function T:\mathbb{R}^3 -> \mathbb{R}^3 defined by:

    T(x) = b x x for x \in \mathbb{R}^3,

    where b x x is the cross product, a linear map? Prove your answer. Find a matrix A which transforms the vector x into its function T(x) = (x')


    I made a complete mess of this. I tried to prove it using:

    T(\lambda_1 x + \lambda_2 y) = \lambda_1 T(x) + \lambda_2 T(y)<br />

    But as you can imagine, this became a massive mess very fast. Is there a particular way to prove this without resorting to performing many cross products? Thanks!
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  2. #2
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    Interesting - T is in fact linear. Do additivity and scalar multiplication seperately using the definition of cross product. The computations are quite easy.
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  3. #3
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    Yeah, I think the method I chose to prove it was too long-winded for a cross product. It gets messy when you're calculating \lambda_1 x + \lambda_2 y crossed with b.
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