Q1: Let $V$ be a $3$-dimensional real vector space with basis $\mathcal{B}$ and $W$ be a $4$-dimensional real vector space with $\mathcal{C}$. What is correct for a linear map $\ell: V \rightarrow W$?

A1: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(4\times 3)$-matrix with real entries.

A2: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(3\times 4)$-matrix with real entries.

A3: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(3\times 3)$-matrix with complex entries.

A4: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(4\times 4)$-matrix with complex entries.

Q2: Can the matrix $\begin{pmatrix} 1 & 0 & 3 \ 2 & 1 & 1 \end{pmatrix}$ be a matrix representation of the linear map $ \ell: \mathcal{P}_2(\mathbb{R}) \rightarrow \mathcal{P}_1(\mathbb{R}) $ given by $\ell(p) = p^\prime$.

A1: Yes, if one chooses suitable bases $\mathcal{B}$ and $\mathcal{C}$.

A2: No, the matrix is of wrong size.

A3: No, because the matrix has an entry given by 3.

A4: No, the matrix has only real entries.