Q1: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear map given by $$f(x,y) = \begin{pmatrix} 2y + 3 x \ 5 x - y \end{pmatrix}$$ What is the corresponding matrix $A$ such that $$f(x,y) = A \binom{x}{y}$$ holds?

A1: $$ A = \begin{pmatrix} 3 & 2 \ 5 & -1 \end{pmatrix} $$

A2: $$ A = \begin{pmatrix} 3 & -2 \ -5 & -1 \end{pmatrix} $$.

A3: $$ A = \begin{pmatrix} 2 & 3 \ 5 & -1 \end{pmatrix} $$

A4: $$ A = \begin{pmatrix} 2 & 3 \ 5 & 1 \end{pmatrix} $$

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear map. Is it possible that $f_A = f_B$ for different matrices $A,B \in \mathbb{R}^{2 \times 2}$.

A1: No, never.

A2: Yes, always.

A3: Only if $f = 0$.

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a linear map. What is the first column of the matrix $A$ such that $f = f_A$?

A1: $f(e_1)$

A2: $f(e_2)$

A3: $f(e_1 + e_2)$

A4: One needs more information.