For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis $A=\{1,t,...,t^d\}$ and $B=\{1,(t-a),...,(t-a)^d\}$.
I know, that i have to write the $t^i$ as a linear combination of $(t-a)^j$. So for
$t^0 = 1*(t-a)^0$
$t^1 = a*(t-a)^0 + 1*(t-a)^1$
$t^2 = (-a^2 + 2a^2) * (t-a)^0 + 2a*(t-a)^1 + 1*(t-a)^2$
$t^3= a^3(t-a)^0 -a^2(t-a)^1+a(t-a)^2+1(t-a)^3$
$...$
How can i figure out a system for the general case?
So the matrix representation is \begin{pmatrix} 1 & 0 & 0 & 0 & 0 &\dots & 0\\ a & 1 & 0 & 0 & 0 &\dots & 0\\ a^2 & \binom{2}{1} a^{2-1} & 1 & 0 & 0 &\dots & 0\\ \vdots & \dots & & \ddots & \vdots\\ a^d & \binom{2}{1} a^{d-1} &\dots &\binom{d}{i} a^{d-i} &\dots & &1 \end{pmatrix} ? Can someone please tell me, if this is correct?
Do the binomial expansion $$ (z + a)^{k} = a^{k} + \binom{k}{1} a^{k-1} z + \dots + \binom{k}{i} a^{k-i} z^{i} + \dots + z^{k}, $$ and then substitute $t = z + a$.