Linear algebra Matrix with respect to basis

[Technique101]

Homework Statement

Find the matrix of the linear operator with respect to the given basis B.

D: P₂ -> P₂ defined by D(ax² + bx + c) = 2ax+b, B = { 3x²+2x+1, x²-2x, x²+x+1 }

Homework Equations

None.

The Attempt at a Solution

I set the basis B = { (3,2,1), (1,-2,0), (1,1,1) } based on the equations

then I did
D(3,2,1) = 6x+2 = (0,6,2)
D(1,-2,0) = 2x-2 = (0,2,-2)
D(1,1,1) = 2x-1 = (0,2,-1)

I believe those are the steps I have to take? I'm just not sure where to go from here.

Thanks!

 

[lanedance]

I would probably start with the D matrix in the {x^2,x,1} basis, then use the B matrix you found to transform it into the new basis

 

[lanedance]

now for clarity let the bases be given by
$$e = {e_1, e_2, e_3} = {x^2,x,1}$$

$$b = {b_1, b_2, b_3} = { 3x2+2x+1, x2-2x, x2+x+1 }$$

so as another way, you have the effect of the matrix D on the b basis vectors, you could re-write the resultant vectors in terms of the b basis to find D relative to b (call it D^b)

though the first method outlined is probably better

 

[Technique101]

Hmm, I don't know if I quite follow. So:

D(1,0,0) = 2x = (0,2,0)
D(0,1,0) = 1 = (0,0,1)
D(0,0,1) = 0 = (0,0,0)

So what would be the next step?

 

[lanedance]

ok, so given a column vector in the e basis u^e = (a,b,c)^T representing (ax^2+b+c), what is the matrix D^e, such that result is v^e in also the e basis

$$v^e = D^e.u^e$$

you've got all the values, just put them in matrix form
D^e =
[ ? ? ?]
[ ? ? ?]
[ ? ? ?]

the you need to find the matrix T, that transforms from the b basis to the v basis, so given a vector u^b, what matrix T, takes it to u^e

$$u^e = T.u^b$$

u^b & u^e are the same vector just written in different bases. The matrix T will be very closely related to the set of vectors B you gave.