# Find the matrix representation relative to the standard bases

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 $$C^3$$ ?

• Welcome to MathSE. Can you show your attempts for to solve this problem? Dec 17 '20 at 16:10
• 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, $[T] = \begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}$ Dec 17 '20 at 16:17
• Ok, please add that attempt in your post. Also, what's mean $C$? Is $C$ the complex field? Dec 17 '20 at 16:27
• yes complex field Dec 17 '20 at 16:30

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