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Linear Algebra - Show that T is Linear

  1. Sep 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Let y [itex]\in[/itex]ℝ[itex]^{3}[/itex] be a fixed vector, and define T:ℝ[itex]^{3}[/itex]→ℝ[itex]^{3}[/itex] to be Tx = X [itex]\times[/itex] Y, the cross product of x and y.
    Show that T is linear.


    2. Relevant equations



    3. The attempt at a solution
    For this question do we have to define another T with the cross product of two other variables to prove this?

    For example:

    Tx = x [itex]\times[/itex] y
    Ta = a [itex]\times[/itex] b

    and we can prove linearity by T(cx+da) = cTx + dTa ?

    Solution in this problem is welcome, since this assignment is already overdue.

    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited by a moderator: Sep 25, 2013
  2. jcsd
  3. Sep 25, 2013 #2

    Mark44

    Staff: Mentor

    No, Ta = a X y
    Yes.
    On-time or overdue, PF policy is that we don't do the work for you. We'll help you with it, though.
     
  4. Sep 25, 2013 #3
    would T(cx+da) be something like:

    cx1 + da1.

    cx2 + da2. x(cross product) [y1,y2,y3]?

    cx3 + da3.
     
  5. Sep 25, 2013 #4

    Office_Shredder

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    Staff Emeritus
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    Gold Member

    Assuming that you have a column vector on the left and a row vector on the right, that is exactly what it would look like.

    You can save yourself a bit of notational pain by doing this in two steps: T is linear if T(x+a) = T(x) + T(a) for all vectors x and a, and T(cx) = cT(x) for all vectors x and scalars c. This way you don't have to worry about keeping track of c's and d's when doing the hard part (showing additive linearity)
     
  6. Sep 25, 2013 #5
    Thanks for your helps.
     
  7. Sep 25, 2013 #6

    Mark44

    Staff: Mentor

    Office_Shredder, I think you have a misconception here. This transformation performs the cross product (not matrix product) of its argument and some fixed vector in R3. For the ordinary cross product, all you need are two vectors in R3.
     
  8. Sep 25, 2013 #7

    Office_Shredder

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    OK I agree there was no reason to write it in that particular format, I simply meant that it wasn't 100% clear to me if he was trying to write two vectors being cross-producted with each other.
     
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