Finding Coordinate Matrix for Linear Transformation T

[Rifscape]

Homework Statement

Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question.

Consider the linear transformation T from
V = P2
to
W = P2
given by
T(a0 + a1t + a2t^2) = (2a0 + 4a1 + 2a2) + (2a0 + 3a1 + 4a2)t + (−2a0 + 3a1 + 4a2)t^2
Let F = (f1, f2, f3) be the ordered basis in P2 given by f1(t) = 1, f2(t) = 1 + t, f3(t) = 1 + t + t^2
Find the coordinate matrix [T]FF
of T relative to the ordered basis F used in both V and W, that is, fill in the blanks below: (Any entry that is a fraction should be entered as a proper fraction, i.e. as either x/y or -x/y where x and y are positive integers with no factors in common.)
T(f1(t)) = _f1(t) + _f2(t) + _f3(t)
T(f2(t)) = _f1(t) + _f2(t) + _f3(t)
T(f3(t)) = f1(t) + f2(t) + f3(t)
and therefore :
[T]FF
= the matrix created by the coefficients of the answers above

Homework Equations

Basis equation.

The Attempt at a Solution

I tried doing what I did for the previous question. Since f1(t) = 1. It is 1 + 0t + 0t^2 = 1. So when I plugged in this to the T(a0 + a1t + a2t^2) I got 2 + 2t - 2t^2 for the coefficients of the first matrix. However when I enter this answer in, it is wrong except for the -2 coefficient for t^2. I have no idea why. I think it has something to do with how f2(t) is 1 + t, and how f2(t) is 1 + t + t^2. Could someone please help me on this?

Thank you for your time.

 

[Mark44]

Homework Statement

Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question.

Consider the linear transformation T from
V = P2
to
W = P2
given by
T(a0 + a1t + a2t^2) = (2a0 + 4a1 + 2a2) + (2a0 + 3a1 + 4a2)t + (−2a0 + 3a1 + 4a2)t^2
Let F = (f1, f2, f3) be the ordered basis in P2 given by f1(t) = 1, f2(t) = 1 + t, f3(t) = 1 + t + t^2
Find the coordinate matrix [T]FF
of T relative to the ordered basis F used in both V and W, that is, fill in the blanks below: (Any entry that is a fraction should be entered as a proper fraction, i.e. as either x/y or -x/y where x and y are positive integers with no factors in common.)
T(f1(t)) = _f1(t) + _f2(t) + _f3(t)
T(f2(t)) = _f1(t) + _f2(t) + _f3(t)
T(f3(t)) = f1(t) + f2(t) + f3(t)
and therefore :
[T]FF
= the matrix created by the coefficients of the answers above

Homework Equations

Basis equation.

The Attempt at a Solution

I tried doing what I did for the previous question. Since f1(t) = 1. It is 1 + 0t + 0t^2 = 1. So when I plugged in this to the T(a0 + a1t + a2t^2) I got 2 + 2t - 2t^2 for the coefficients of the first matrix.

This is not the "first matrix." This is $T(f_1)$, the first column of a matrix representation of T. Now calculate $T(f_2)$ and $T(f_3)$, which will be, respectively, the 2nd and 3rd columns of your matrix.

This matrix won't be the answer, as those three column vectors are in terms of the standard basis ({1, t, t²}), not the basis of this problem.

However when I enter this answer in, it is wrong except for the -2 coefficient for t^2. I have no idea why. I think it has something to do with how f2(t) is 1 + t, and how f2(t) is 1 + t + t^2. Could someone please help me on this?

Thank you for your time.