Q1: Which of the following claims is correct?

A1: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is a linear combination of the columns of $A$.

A2: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^m $, the vector $A \mathbf{x}$ is well-defined.

A3: For $ A \in \mathbb{R}^{n \times m} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is well-defined.

A4: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is a linear combination of the rows of $A$.

Q2: Which of the following equations is correct?

A1: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = x_1 \begin{pmatrix} 2 \ 0 \end{pmatrix} + x_2 \begin{pmatrix} 1 \ 1 \end{pmatrix} $$

A2: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = x_1 \begin{pmatrix} 2 \ 1 \end{pmatrix} + x_2 \begin{pmatrix} 1 \ 0 \end{pmatrix} $$

A3: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = 0 $$

A4: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = \begin{pmatrix}x_1 \ x_2 \end{pmatrix} $$

A5: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = \begin{pmatrix} 2 x_1 \ x_2 \end{pmatrix} $$

Q3: A matrix $A \in \mathbb{R}^{m \times n}$ defines a map $f_A$. What is the correct definition?

A1: $ f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m , , ~ \mathbf{x} \mapsto A \mathbf{x}$

A2: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A \mathbf{x}$

A3: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A - \mathbf{x}$

A4: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A + \mathbf{x}$

A5: $ f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m , , ~ \mathbf{x} \mapsto \mathbf{x}$