Find the matrix representations of the Differentiation Map in the Basis

[nautolian]

Homework Statement

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.

Homework Equations

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

 

[jambaugh]

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.

 

[jambaugh]

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.