We have linear transformation $F = \mathbb{P_{2}} \rightarrow \mathbb{R^{2}} $, where $ F(p(t)) = \begin{pmatrix} p(0) \\ P(1) \end{pmatrix},$ $ A = \{1,t,t^{2}\},$ $B=\left \{ \begin{pmatrix} 1\\ 0\end{pmatrix},\begin{pmatrix} 0\\ 1\end{pmatrix} \right \}$.
From this, transformation matrix is $$M_B^A=\begin{pmatrix}{1}&{0}&{0}\\{1}&{1}&{1}\end{pmatrix}.$$
How should I calculate now $w = F(v)$ direct and also over the relation $c_{B}(w) = A\cdot c_{A}(v)$, when $v = a+bt+ct^{2}$?
By definition, we have
$$ F(v) = \begin{pmatrix} v(0) \\ v(1) \end{pmatrix} = \begin{pmatrix} a \\ a + b + c \end{pmatrix}. $$
Since $v = a \cdot 1 + b \cdot t + c \cdot t^2$, we also have
$$ c_A(v) = \begin{pmatrix} a \\ b \\ c \end{pmatrix} $$
and so
$$ c_B(F(v)) = M^A_B c_A(v) = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} a \\ a + b + c \end{pmatrix} $$
and indeed
$$ c_B(F(v)) = c_B \left( \begin{pmatrix} a \\ a + b + c \end{pmatrix} \right) = \begin{pmatrix} a \\ a + b + c \end{pmatrix} $$
as $B$ is the standard basis of $\mathbb{R}^2$.