Linear Algebra matrix linear transformation

[Rifscape]

Homework Statement

Consider the linear transformation T from
V = P2
to
W = P2
given by
T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2
Let E = (e1, e2, e3) be the ordered basis in P2 given by
e1(t) = 1, e2(t) = t, e3(t) = t^2
Find the coordinate matrix [T]EE of T relative to the ordered basis E 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(e1(t)) = _e1(t) + _ e2(t) + _e3(t)
T(e2(t)) = _e1(t) + _e2(t) + _e3(t)
T(e3(t)) = _e1(t) + _e2(t) + _e3(t)
and therefore :
[T]EE = the combined matrix of the coefficients of the above three equations.

Sorry for the poor formatting

Homework Equations

No idea

The Attempt at a Solution

I have no idea
I have no idea how to do this, if someone can give me a step by step solution so that I can understand each step and process I would appreciate it greatly.
Thanks for your help.

 

[HallsofIvy]

Homework Statement

Consider the linear transformation T from
V = P2
to
W = P2
given by
T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2

You mean T(a0+ a1t+ a2t^2) on the left, right?

Let E = (e1, e2, e3) be the ordered basis in P2 given by
e1(t) = 1, e2(t) = t, e3(t) = t^2
Find the coordinate matrix [T]EE of T relative to the ordered basis E 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(e1(t)) = _e1(t) + _ e2(t) + _e3(t)
T(e2(t)) = _e1(t) + _e2(t) + _e3(t)
T(e3(t)) = _e1(t) + _e2(t) + _e3(t)
and therefore :
[T]EE = the combined matrix of the coefficients of the above three equations.

Sorry for the poor formatting

Homework Equations

No idea

The Attempt at a Solution

I have no idea
I have no idea how to do this, if someone can give me a step by step solution so that I can understand each step and process I would appreciate it greatly.
Thanks for your help.

To find the matrix corresponding to a linear transformation in a given ordered basis, apply that linear transformation to each basis vector in turn and write the result as a linear combination of those basis vectors. The coefficients give a column of the basis. For example, to apply the linear transformation two the first basis vector, T(e1), note that e1= 1+ 0t+ 0t^2. That is, a0= 1, a1= 0, and a2= 0. Since T is defined by "T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2". T(1+ 0t+ 0t^2)= (-4(1)+ 2(0)+ 3(0))+ (2(1)+ 3(0)+ 3(0))t+ (-2(1)+ 4(0)+ 3(0))t^2= -4+ 2t- 2t^2. The first column of the matrix is \begin{pmatrix}-4 \\ 2 \\ -2\end{pmatrix}.

Now do the same to the second and third basis vectors.

 

[Rifscape]

T(f1(t)) = f1(t) + f2(t) + f3(t)
When I put in the first one which is f1(t) = 1, so it would be 1 + 0t + 0t^2. So when I plug it in, I should get (2(1) + 0 + 0) + (2(1) + 0 + 0)t + (-2(1) + 0 + 0)t^2
= (2,2,-2) However when I put this in it is wrong, does it have something to do with how f2(t) is now 1 + t, and f3(t) is now 1 + t + t^2?
T(f2(t)) = f1(t) + f2(t) + f3(t)
T(f3(t)) = f1(t) + f2(t) + f3(t)

 

[Ray Vickson]

T(f1(t)) = f1(t) + f2(t) + f3(t)
When I put in the first one which is f1(t) = 1, so it would be 1 + 0t + 0t^2. So when I plug it in, I should get (2(1) + 0 + 0) + (2(1) + 0 + 0)t + (-2(1) + 0 + 0)t^2
= (2,2,-2) However when I put this in it is wrong, does it have something to do with how f2(t) is now 1 + t, and f3(t) is now 1 + t + t^2?
T(f2(t)) = f1(t) + f2(t) + f3(t)
T(f3(t)) = f1(t) + f2(t) + f3(t)

So, you have $T(e_1) = 2 e_1 + 2 e_2 - 2 e_3$. Why do you think this is wrong?

 

[Rifscape]

When I enter the numbers in it says it is wrong, except the - 2 at the end which is right. I feel like I am doing something wrong since f2(t) is now 1+t.

 

[Mark44]

T(f1(t)) = f1(t) + f2(t) + f3(t)
When I put in the first one which is f1(t) = 1, so it would be 1 + 0t + 0t^2. So when I plug it in, I should get (2(1) + 0 + 0) + (2(1) + 0 + 0)t + (-2(1) + 0 + 0)t^2
= (2,2,-2) However when I put this in it is wrong, does it have something to do with how f2(t) is now 1 + t, and f3(t) is now 1 + t + t^2?
T(f2(t)) = f1(t) + f2(t) + f3(t)
T(f3(t)) = f1(t) + f2(t) + f3(t)

Where did f1, f2, and f3 come from? The same basis, i.e., $e_1, e_2,$ and $e_3$ is used in both V and W. In this problem V and W are the same spaces.
You aren't using the definition of the transformation, which I've copied below.
$T(a_0 + a_1t + a_2t^2) = (−4a_0 + 2a_1 + 3a_2) + (2a_0 + 3a_1 + 3a_2)t + (−2a_0 + 4a_1 + 3a_2)t^2$
Use this definition to see what T does to $e_1$ (HallsOfIvy already did this one), $e_2$, and $e_3$.

 

[Rifscape]

No sorry I didn't clarify. I finished the last question, this is a completely new question. I solved the previous question with hall of ivys help. However when I apply the same strategy to this question I cannot solve it. This question has f instead of e. Sorry for the confusion

 

[Mark44]

No sorry I didn't clarify. I finished the last question, this is a completely new question.
I solved the previous question with hall of ivys help. However when I apply the same strategy to this question I cannot solve it. This question has f instead of e. Sorry for the confusion

Then you should start a new thread, with the complete problem statement of the new problem.