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Find the matrix representations of the Differentiation Map in the Basis

  1. Oct 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B.

    2. Relevant equations



    3. The attempt at a solution

    I have proved that it is a linear transformation, but I'm really not sure where to begin with the second part. Any help would be appreciated. Thanks
     
  2. jcsd
  3. Oct 15, 2012 #2

    jambaugh

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    I take it St means "standard basis" I.e. {1,x,x^2}.
    First see what the action of D is on the basis vectors for each basis and express these results as a linear combination of basis elements. Then see how the matrix form is supposed to do exactly the same thing to match up components.
     
  4. Oct 15, 2012 #3

    jambaugh

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    I find it helpful to express expansion in a given basis by writing the row (matrix) of basis elements and the element as a matrix product. Thus e.g.

    [tex] ax^2 + bx + c = \left[ 1, x, x^2 \right]\cdot \left(\begin{array}{c}c \\ b\\ a\\ \end{array}\right) [/tex]

    You can then solve for the components of the matrix [itex]\mathbf{M}[/itex] in the equation:

    [tex]D \left\{ \left[ \mathbf{basis}_1 \right] \cdot\left(\begin{array}{c}c \\ b\\ a\\ \end{array}\right) \right\}=\left[\mathbf{basis}_2\right] \cdot \left\{ \mathbf{M}\cdot \left(\begin{array}{c}c \\ b\\ a\\ \end{array}\right)\right\} [/tex]
    EDIT: Oops! I had my products backward, now fixed.
     
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