I'm stuck at this problem:
Let $P_3(\mathbb{C})$ denote the complex vector space of complex polynomials of degree 2 or less. Let $\alpha,\beta\in\mathbb{C},\alpha\neq\beta$. Consider the image $L:P_3(\mathbb{C})\rightarrow \mathbb{C}^2$ given by $L(p)=\begin{pmatrix}p(\alpha)\\ p(\beta)\end{pmatrix}\text{for} \: p\in P_3(\mathbb{C})$. Consider again the basis $\mathcal{V}=(1,X,X^2)$ for $P_3(\mathbb{C})$ and the standard basis $\mathcal{E}=(\pmb{e_1},\pmb{e_2})$ for $\mathbb{C}^2$.
- Find the matrix representation $_\mathcal{E}[L]_\mathcal{V}$ for $L$ with regards to $\mathcal{V}$ and $\mathcal{E}$.
What do I do? If you have any literature that can help me understand what to do, I will greatly appreciate you would give me a link :S
Best Answer
It has to be a $2 \times 3$ matrix, since $P_3$ is 3-dimensional, and $\Bbb{C}^2$ is 2-dimensional. The first column should be the image of the first basis vector of $P_3$, which is $p(x) = 1$. Regardless of $\alpha$ and $\beta$, you get $L(1) = \binom{1}{1}$.
Now, for the second basis vector, $p(x) = x$, the image of $L$ is $L(x) = \binom{\alpha}{\beta}$.
Finally, for the third, $p(x) = x^2$, and we get $L(x^2) = \binom{\alpha^2}{\beta^2}$.
So the matrix is given by
$$ \left( \begin{array}{cc} 1 & \alpha & \alpha^2 \\ 1 & \beta & \beta^2 \end{array} \right) $$