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Homework Help: Change of basis

  1. Oct 22, 2007 #1
    Hey i was just doping someone wouldnt mind looking over my working to see if im on the right track!
    *T(x,y,z)=(-x-y-z,x+y-5z,-3x-3y+3z) is a linear transformation.
    S is the standard basis, S={e1,e2,e3} and B is another basis, B={v1,v2,v3} where:
    e1=(1,0,0) e2=(0,1,0) e3=(0,0,1) v1=(1,1,1,) v2=(1,-1,0) v3=(0,1,-1)
    - [T]S->S = [1 0 0
    0 1 0
    0 0 1]
    -P B->S = [1 1 0
    1 -1 1
    1 0 -1]
    -P S->B = [1/3 1/3 1/3
    2/3 -1/3 -1/3
    1/3 1/3 -2/3]

    -[e2]B = P S->B.[e2]S
    = (1/3,-1/3,1/3)
    -[T(e2)]B =? what does this refer to? Do I have to refer to the equation in any part of these? as in the matrix [-1 -1 -1
    1 1 -5
    -3 -3 3]
    Any help is greatly appreciated!
  2. jcsd
  3. Oct 23, 2007 #2


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    Homework Helper

    What is the problem to be solved?
  4. Oct 24, 2007 #3
    The question is: What is [T(e2)]B?
  5. Oct 24, 2007 #4


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    A good way of finding a matrix form for a linear transformation, in a given basis, is to apply the transformation to each of the basis vectors, in turn, and write the result in terms of the given basis. Each of those will be one column of the matrix.
    For example, what do you get if you apply this transformation to v1= (1, 1, 1)? Now write that result as av1+ bv2+ cv3. The numbers a, b, c will be the first column of the matrix.

    (This problem has obviously been set up to make it easy to do that. Applying the transformation to v2 is particularly interesting.)
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