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Linear transformation and Change of Basis

  1. Jul 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance.

    If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and
    T(v2)=3v2.

    and B = v1=(1,1) , v2=(1,-1)


    Find T with respect to the base B and T with respect to Nat, (the Natural Base)

    2. Relevant equations



    3. The attempt at a solution

    I found T with respect to B by inspection

    -1 0
    -2 3

    How can I find T with respect to the natural base?

    Thanks
     
  2. jcsd
  3. Jul 23, 2012 #2
    Write the two natural basis vectors in terms of [itex]v_1,v_2[/itex] and then see what this transformation does to them. Once you know that, then you know how to find the matrix.
     
  4. Jul 24, 2012 #3
    Do you mean to write 1,0 and 0,1 as a linear combination of v1 and v2 ? how can I see what the transformation does to them when I'm not given the transformation?

    Thanks
     
  5. Jul 24, 2012 #4
    Well, let's say [itex]e_1 = c_1v_1 + c_2v_2[/itex]. Then, [itex]T(e_1) = T(c_1v_1 + c_2v_2)[/itex]. Now, use the linearity of [itex]T[/itex] and what you know about [itex]T(v_1)[/itex] and [itex]T(v_2)[/itex] to calculate [itex]T(e_1)[/itex].

    EDIT:
    I don't know what terms your book uses, but I mean that [itex]e_1 = (1,0)[/itex].
     
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