# Matrix representation of linear transformation

I'm having trouble with part of homework exercise. Let $\mathcal{P}_3(\mathbb{C})$ be the vector space of complex polynomials of degree 2 or lower and $\alpha,\beta\in\mathbb{C},\alpha\ne \beta$.

Let $L:\mathcal{P}_3(\mathbb{C})\to\mathbb{C}^2$ given by $L(p)=(p(\alpha), p(\beta))^T$ and I want to find the matrix representation of $L$ with respect to the bases $\{1,X,X^2\}$ and $\{\mathbf{e_1},\mathbf{e_2}\}$.

I usually apply $L$ to the elements of the basis for $\mathcal{P}_3$, but with this definition of $L$ I don't know how to do it. Can someone give me a pointer?

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$$L(1):=\binom{1(\alpha)}{1(\beta)}=\binom{1}{1}=1\cdot\binom{1}{0}+1\cdot\binom{0}{1}$$
$$L(x):=\binom{x(\alpha)}{x(\beta)}=\binom{\alpha}{\beta}=\alpha\cdot\binom{1}{0}+\beta\cdot\binom{0}{1}$$
$$L(x^2):=\binom{x^2(\alpha)}{x^2(\beta)}=\binom{\alpha^2}{\beta^2}=\alpha^2\cdot\binom{1}{0}+\beta^2\cdot\binom{0}{1}$$
So the matrix representation is $\begin{pmatrix}1 & \alpha & \alpha^2\\ 1 & \beta & \beta^2\end{pmatrix}$? – user37158 Mar 7 '13 at 21:46