I'm solving a linear algebra problem. I have linear transformation $D$:
$D : R_2[t] \rightarrow R_2[t]$
$D(p) = \frac{d}{dt}p$
and bases:
$A = \{1 + t, 1- t, t^2\}$
$B = \{1 + t, 1 - t\}$
Now I need to discover matrix of linear transformation $D$ from $A$ to $B$.
Well, I started up by writing down a typical polynom in canonical base:
$p(t) = a + bt + ct^2$
Then I tried to discover what would be its representation in base $A$, by doing:
$a + bt +ct^2 = x(1 + t) + y(1 - t) +zt^2$
So,
$[p(t)]_A = \left(\frac{a+b}{2}, \frac{a-b}{2}, c\right)$
Good. Now, I know that:
$D(p(t)) = b + 2c$
Then:
$[D(p(t))] = (b, 2c) = \begin{bmatrix} b \\ 2c \\ \end{bmatrix} $
Then
$ [D(p(t))] = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix} \begin{bmatrix} a \\ b \\ c \\ \end{bmatrix}$
So,
$ [D(p(t))]_A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix} \begin{bmatrix} \frac{a+b}{2} \\ \frac{a-b}{2} \\ c \\ \end{bmatrix}$
Ok is clear not the transformation linear from $A$ to $B$, but only this representation from the canonical base to $A$, only? What should I do? Thanks.