**Question:**Let $(1, z, z^2)$ be the standard basis for $\mathbb{C}^2$. Let $(1, z - 1, (z - 1)^2$ be another basis for $\mathbb{C}^2$. Find a matrix transformation from $p(z) = a_0 + a_1z + a_2z^2$ to the polynomial $q(z) = b_0 + b_1(z - 1) + b_2(z - 1)^2$.
My idea so far is to expand $q$, viz.,
$$q(z) = (b_0 - b_1 + b_2) + (b_1 - 2b_2)z + b_2z^2.$$
Then, I get confused. My attempt was to look at a linear transformation $T$ from $p$ to $q$ by considering the action of $(1, z, z^2)$ on $T$, but this didn't get me anywhere.
My next attempt was to consider a change of basis matrix, but I am struggling on implementing the idea. Any help?
I think the question is confusing and poorly-worded. It asks,
The problem is, what is the transformation? What do $p(z)$ and $q(z)$ have to do with each other? And what do their coefficients have to do with each other?
If we assume that $a_i = b_i$ for $i = 0, 1, 2$, then we get a perfectly valid transformation, but in finding a matrix for this transformation, which bases should we consider? If we consider from $(1, z, z^2)$ to $(1, z - 1, (z - 1)^2)$ (as one might guess), then the corresponding matrix is the identity!
I think the more likely guess is that $p(z)$ is supposed to equal $q(z)$, and that you were always supposed to compute a change-of-basis from $(1, z, z^2)$ to $(1, z - 1, (z-1)^2)$. This is confusing, given that the question asks for a transformation, mapping one polynomial to another. When finding the change of basis matrix, the underlying transformation is the identity transformation; it doesn't change the polynomial at all, just the coordinates used to represent it.
Anyway, assuming the latter guess is the correct interpretation, we compute the matrix for the identity transformation from $(1, z, z^2)$ to $(1, z - 1, (z - 1)^2)$. We first transform all the vectors in the first basis: \begin{align*} I(1) &= 1 \\ I(z) &= z \\ I(z^2) &= z^2. \end{align*} (No trick; applying the identity transformation is super easy!)
We then take these results and write them as (unique) linear combinations of $(1, z - 1, (z - 1)^2)$. This is the slightly trickier bit. We have: \begin{align*} I(1) &= 1 = 1 \cdot 1 + 0 \cdot (z - 1) + 0 \cdot (z - 1)^2 \\ I(z) &= z = 1 \cdot 1 + 1 \cdot (z - 1) + 0 \cdot (z - 1)^2 \\ I(z^2) &= z^2 = 1 \cdot 1 + 2 \cdot (z - 1) + 1 \cdot (z - 1)^2. \end{align*} Letting $B$ be the second basis, and reading off the coordinates, as column vectors, we have: $$[I(1)]_B = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \quad [I(z)]_B = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \quad [I(z^2)]_B = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}.$$ Thus, our change of basis matrix becomes $$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}.$$ And that's my best guess as to what the question wants. Hope it helps.