Linear Algebra - Change of basis question

[zeion]

1. The problem statement, all variables and given/known data

Let A = E4 in R4 (standard basis) and B = {x^2, x, 1} in P2 over R. If T is the linear transformation that is represented by


\begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix}



relative to A and B, find the matrix that represents T with respect to A' and B' where
A' = {(1,0,0,0), (0,0,1,0), (1,-1,0,0), (0,-1,1,1)}
B' = {x^2 + 1, x, 1}

2. Relevant equations



3. The attempt at a solution

So by looking at this matrix T, it's clear that its a transformation from A to B, so we want the transformation matrix T_{B'A'},
which is: T_{B'A'} = I_{B'B}T_{BA}I_{AA'}

So I need to find I_{AA'} and I_{B'B}.

For I_{AA'}, I write A' wrt A(which is standard basis of R4):

I get : I_{AA'} = \begin{bmatrix}1 & 0 & 1 & 0 \\0 & 0 & -1 &-1\\0 & 1 & 0 &1\\ 0&0&0&1 \end{bmatrix}

Then for I_{B'B}, I write B wrt B', and get

I_{B'B} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\-1 & 0 & 1 \end{bmatrix}

Now I put them together to get something with lots of zeros.. which doesn't seem right?

[vela]

Your answer is correct. (At least it sounds correct.) You have the representations of the A' basis in the natural basis, so try applying them to the given T and see what you get. Then look at how those results would be represented in the B' basis. You'll see why the transformed T has so many zeros.

[zeion]

When I put them all together I get:

\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\-1 & 0 & 1 \end{bmatrix} \begin{bmatrix}1 & 1 & 0 & 1\\0 & 0 & 1 & -1\\1 & 1 & 0 & 1 \end{bmatrix} \begin{bmatrix}1 & 0 & 1 & 0 \\0 & 0 & -1 &-1\\0 & 1 & 0 &1\\ 0&0&0&1 \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 0 & 0 \end{bmatrix}
??