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Answer:
Step-by-step explanation:
First we start by finding the dimension of the matrix [T]EE
The dimension is : Dim (W) x Dim (V) = 3 x 3
Because the dimension of P2 is the number of vectors in any basis of P2 and that number is 3
Then, we are looking for a 3 x 3 matrix.
To find [T]EE we must transform the vectors of the basis E and then that result express it in terms of basis E using coordinates and putting them into columns. The order in which we transform the vectors of basis E is very important.
The first vector of basis E is e1(t) = 1
We calculate T[e1(t)] = T(1)
In the equation : 1 = a0
And that is the first column of [T]EE
The second vector of basis E is e2(t) = t
We calculate T[e2(t)] = T(t)
in the equation : 1 = a1
Finally, the third vector of basis E is
in the equation : a2 = 1
Then
And that is the third column of [T]EE
Let's write our matrix
T(X) = AX
Where T(X) is to apply the transformation T to a vector of P2,A is the matrix [T]EE and X is the vector of coordinates in basis E of a vector from P2
For example, if X is the vector of coordinates from e1(t) = 1
Applying the coordinates 2,6 and -2 to the basis E we obtain
That was the original result of T[e1(t)]