Question

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.)

Answer 1

`[T]EE=\left[\begin{array}{ccc}2&3&3\\6&4&4\\-2&3&4\end{array}\right]`

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

`T(1)=(2.1+3.0+3.0)+(6.1+4.0+4.0)t+(-2.1+3.0+4.0)t^(2)=2+6t-2t^(2)`

`[T(e1)]E=\left[\begin{array}{c}2&6&-2\\\end{array}\right]`

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

`T(t)=(2.0+3.1+3.0)+(6.0+4.1+4.0)t+(-2.0+3.1+4.0)t^(2)=3+4t+3t^(2)`

`[T(e2)]E=\left[\begin{array}{c}3&4&3\\\end{array}\right]`

Finally, the third vector of basis E is
`e3(t)=t^(2)`

`T[e3(t)]=T(t^(2))`

in the equation : a2 = 1

`T(t^(2))=(2.0+3.0+3.1)+(6.0+4.0+4.1)t+(-2.0+3.0+4.1)t^(2)=3+4t+4t^(2)`

Then

`[T(t^(2))]E=\left[\begin{array}{c}3&4&4\\\end{array}\right]`

And that is the third column of [T]EE

Let's write our matrix

`[T]EE=\left[\begin{array}{ccc}2&3&3\\6&4&4\\-2&3&4\end{array}\right]`

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

`X=\left[\begin{array}{c}1&0&0\\\end{array}\right]`

`AX=\left[\begin{array}{ccc}2&3&3\\6&4&4\\-2&3&4\end{array}\right]\left[\begin{array}{c}1&0&0\\\end{array}\right]=\left[\begin{array}{c}2&6&-2\\\end{array}\right]`

Applying the coordinates 2,6 and -2 to the basis E we obtain

`2+6t-2t^(2)`

That was the original result of T[e1(t)]