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No idea where to start for linear transformation question from P2->P2

  1. Jul 22, 2011 #1
    Let T:P2[itex]\rightarrow[/itex]P2 be given by

    T(x-1)=1-x
    T(x2-2x)=-1+x-2x2
    T(3-x2)=-1+2x+3x2

    Find the matrix for T with respect to the standard basis B={1,x,x2}for P2

    To be honest, I have no idea where to start. Help would be greatly appreciated
     
  2. jcsd
  3. Jul 23, 2011 #2

    ehild

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    Homework Helper
    Gold Member

    Write the polynomials as column vectors like
    x-1 --->
    -1,
    1
    0
    x^2-2x --->
    0
    -2
    1

    and so on and make a matrix A from the three columns on the left hand side of the equation and a matrix B from the columns at the right-hand side. You get a matrix equation TA=B which you can solve for T
    by multiplying both sides with the inverse of A:
    T=TA A-1=BA-1.

    ehild
     
  4. Jul 23, 2011 #3
    i dont understand how you got the column vectors
     
  5. Jul 23, 2011 #4

    ehild

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    Gold Member

    1, x, x^2 are the basis, consider them as base vectors e1, e2, e3. The coefficient of x^k in the polynomial is considered as the k-th component.

    For example, -1+x-2x^2 = -1 (1) +1 (x) -2 (x^2) or -1 e1 +1*e2 -2 e3. You can write it as the vector (-1,1,-2), or its transposed as column vector.

    ehild
     
    Last edited: Jul 23, 2011
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