Q1: Consider the linear map $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^2$ be two vectors that have an angle $\alpha$ between them. What is the angle between the images $f_A(\mathbf{u})$ and $f_A(\mathbf{v})$?

A1: It’s also $\alpha$.

A2: It’s $\frac{1}{2}\alpha$.

A3: It’s $\frac{1}{3}\alpha$.

A4: It’s $-\alpha$.

A5: One needs more information.

A6: It’s $0$.

Q2: Consider the linear map $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $A = \begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix} $. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^2$ be two vectors that have an angle $\alpha$ between them. What is the angle between the images $f_A(\mathbf{u})$ and $f_A(\mathbf{v})$?

A1: It’s also $\alpha$.

A2: It’s $\frac{1}{2}\alpha$.

A3: It’s $\frac{1}{3}\alpha$.

A4: It’s $-\alpha$.

A5: One needs more information.

A6: It’s $0$.

Q3: Consider the linear map $f_A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $A = \begin{pmatrix}\cos(\alpha) & -\sin(\alpha) \ \sin(\alpha) & \cos(\alpha) \end{pmatrix} $. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^2$ be two vectors with a right-angle between them. What is the angle between the images $f_A(\mathbf{u})$ and $f_A(\mathbf{v})$?

A1: It’s also a right-angle.

A2: It’s not a right-angle anymore.

A3: One needs more information.