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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 $C^3$ ?

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    $\begingroup$ Welcome to MathSE. Can you show your attempts for to solve this problem? $\endgroup$
    – mathproof
    Dec 17 '20 at 16:10
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    $\begingroup$ 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} $ $\endgroup$
    – Student A
    Dec 17 '20 at 16:17
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    $\begingroup$ Ok, please add that attempt in your post. Also, what's mean $C$? Is $C$ the complex field? $\endgroup$
    – mathproof
    Dec 17 '20 at 16:27
  • $\begingroup$ yes complex field $\endgroup$
    – Student A
    Dec 17 '20 at 16:30
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First, we need to analyze two case:

  1. $\mathbb{C}_{\mathbb{R}}$: Complex space over $\mathbb{R}$.
  2. $\mathbb{C}_{\mathbb{C}}$: Complex space over $\mathbb{C}$.

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 $$[T]=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{pmatrix}$$

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