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Linear transformations

  1. May 10, 2008 #1
    1. The problem statement, all variables and given/known data

    T:[tex]{R^3 \rightarrow {R^2}[/tex] given by [tex]T(v_1,v_2,v_3) = (v_3 -v_1, v_3 - v_2)[/tex]

    If linear, specify the range of T and kernel T

    The attempt at a solution
    Okay, I went ahead and tried to find the kernel of T like here:
    [tex]\begin{align*}&v_3 - v_1 = 0\\
    &v_3 - v_2 = 0\end{align*}[/tex]

    Thus, [tex]\begin{align*}&v_3 = v_1 \\
    &v_3 = v_2\end{align*}[/tex]

    So choosing v3 as s gives the 1-D basis of W= s(1, 1, 1) **a column vector**

    But I'm not entirely sure how to get the range. IF I did the kernel correctly, then that means the dimension of the range will be 2 as 2+1 = 3 (the dimension of the domain). But when I try to do the range, I get a 3-dimensional basis where v1,v2,and v3 are their own LI vectors as so:
    (y1,y2) = s(1,1) + t(-1,0) + r(0,-1)

    Any help?
  2. jcsd
  3. May 11, 2008 #2
    Everything seems fine. So where's your difficulty?
  4. May 11, 2008 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    You have correctly deduced that the range must have dimension 2 and you know that the range is a subspace of R2.

    How many subspaces of dimension 2 do you think R2 has!
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