Q1: Consider the map $f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ given by $f_A(\mathbf{x}) = A \mathbf{x}$ with a matrix $ A \in \mathbb{R}^{m \times n}$. Is the map linear?

A1: Yes, it is always a linear map.

A2: It’s only a linear map for $n = m$.

A3: It’s only well-defined for $n = m$.

A4: No, it’s never a linear map.

A5: One needs more information.

Q2: Consider the maps $f_A: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $f_B: \mathbb{R}^n \rightarrow \mathbb{R}^n$ given by $f_A(\mathbf{x}) = A \mathbf{x}$ and $f_B(\mathbf{x}) = B \mathbf{x}$ with matrices $ A, B \in \mathbb{R}^{n \times n}$. What can one say about the composition $f_A \circ f_B$?

A1: $f_A \circ f_B = f_{AB}$

A2: It’s not well-defined.

A3: $f_A \circ f_B = f_{A+B}$

A4: $f_A \circ f_B = f_{BA}$

A5: $f_A \circ f_B = f_{A}$

Q3: Consider the matrix $A = \begin{pmatrix} 2 & 1 \ 1 & 1\end{pmatrix}$. Which of the following claims is correct?

A1: $ f_A(\mathbf{x}) = \begin{pmatrix} 2x_1 + x_2 \ x_1 + x_2 \end{pmatrix} $

A2: $ f_A(\mathbf{x}) = \begin{pmatrix} 2x_1 - x_2 \ x_1 \cdot x_2 \end{pmatrix} $

A3: $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}$

A4: $f_A: \mathbb{R} \rightarrow \mathbb{R}^2$

A5: $f(\mathbf{x}) = 0$ for all $\mathbf{x} \in \mathbb{R}^2$