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Linear operator problem

  1. Dec 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Let [itex]\phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2}[/itex] be a linear operator defined as: [itex](\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2[/itex] where
    [itex]B=
    \begin{bmatrix}
    3 & -2 \\
    2 & -2 \\
    \end{bmatrix}
    [/itex]

    Find rank,defect and one basis of an image and kernel of linear operator [itex]\phi[/itex].

    2. Relevant equations
    -Linear transformations

    3. The attempt at a solution
    Could someone explain how to find matrix of linear operator [itex]\phi[/itex]?
    Also, is it necessary to know the matrix [itex]A[/itex]?
     
  2. jcsd
  3. Dec 4, 2015 #2

    RUber

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    Start by using A = I, the identity. The linear operator is defined by its effect on the basis elements.
     
  4. Dec 6, 2015 #3
    I have forgotten to mention that [itex]P_2[/itex] is a space of polynomials with degree not larger than [itex]2[/itex].
    Still, I don't understand why matrix [itex]A[/itex] is not given.
     
  5. Dec 6, 2015 #4

    Samy_A

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    A can be any matrix in ##M_{2,2}##.

    Here they tell you how the linear operator ##\phi## transforms any matrix of ##M_{2,2}## into a polynomial.

    As for a tip, see what @RUber suggested.
     
    Last edited: Dec 6, 2015
  6. Dec 6, 2015 #5

    Samy_A

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    Sorry, double posting.
     
  7. Dec 7, 2015 #6

    Ray Vickson

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    The matrix ##A## is not given because it can be any 2 x 2 real matrix, and the definition of ##\phi## tells you how ##A## maps into a quadratic polynomial.

    One fairly straightforward approach would be to take
    [tex] A = \pmatrix{a & b \\ c & d} [/tex]
    and compute the polynomial ##\phi(A)(x)## explicitly.
     
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