Let $T : P_{2}(C) \to C^3$ be a linear transformation such that $$T(a + bx + cx^2) = (a, a + b, a + b + c)$$
Find the matrix representation $[T]$ relative to the standard bases.
If standard basis for $R^3$,
$T(1,0,0)=(1,0,0)$
$T(0,1,0)=(1,1,0)$
$T(0,0,1)=(1,1,1)$
Therefore, $[T]$ = \begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}
But it is the same with $C^3$ ?
First, we need to analyze two case:
Then, the answer dependes if in your problem $T:P_{2}(\mathbb{C}_{\mathbb{C}})\to \mathbb{C}_{\mathbb{C}}^{3}$ or $T:P_{2}(\mathbb{C}_{\mathbb{R}})\to \mathbb{C}_{\mathbb{R}}^{3}$.
Suppose for example the second case, also is well-know that $\mathbb{C}_{\mathbb{R}}\cong \mathbb{R}^{2}$ and $\mathbb{C}_{\mathbb{R}}^{3}\cong \mathbb{R}^{6}$. (In general $\mathbb{C}_{\mathbb{R}}^{n} \cong \mathbb{R}^{2n}$). So, we have that $[T]_{\beta_{1}\to \beta_{2}}$ be a matrix of order $\dim(\mathbb{C}_{\mathbb{R}}^{3})\times \dim(P_{2}(\mathbb{C}_{\mathbb{R}}))$.
Now, in the first case: $T:P_{2}(\mathbb{C}_{\mathbb{C}})\to \mathbb{C}_{\mathbb{C}}^{3}$, we have that your approach is correct. Indeed $$[T]=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{pmatrix}$$
