Consider the linear transformation t from v = p2 to w = p2 given by t(a0 + a1t + a2t2) = (2a0 + 3a1 + 3a2) + (6a0 + 4a1 + 4a2)t + (−2a0 + 3a1 + 4a2)t2 let e = (e1, e2, e3) be the ordered basis in p2 given by e1(t) = 1, e2(t) = t, e3(t) = t2 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.)

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)]

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step-by-step explanation: