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 $V,W$ be $\mathbb{F}$-vector spaces. What is always correct for a linear map $f: V \rightarrow W$?

A1: $ f(0) = 0 $

A2: $ f(x+y) = f(x-y) $

A3: $ f(\lambda \cdot x) = f(x) $

A4: $ f(0\cdot x) = f(x) $

Q3: Let $V = \mathcal{P}(\mathbb{R})$ be the space of real polynomials. Let’s define the map $\ell: V \rightarrow \mathbb{R}$ by $\ell(p) = \int_0^1 p(x) , dx$. Is it linear?

A1: No, because $\ell(0) \neq 0$.

A2: No, becasue $\ell(p+q) \neq \ell(p) + \ell(q)$ in general.

A3: No, because $\ell(x \mapsto x^2) \neq \ell(x \mapsto x) \cdot \ell(x \mapsto x)$

A4: Yes, the two properties are satisfied.