Q1: Let $\mathbf{v},\mathbf{w} \in \mathbb{R}^3$ be given by $\mathbf{v} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} -2 \ 1 \ 0 \end{pmatrix}$. Are the two vectors orthogonal?

A1: Yes!

A2: No!

Q2: Let $\mathbf{v} \in \mathbb{R}^4$ be given by $\mathbf{v} = \begin{pmatrix} 3 \ 4 \ 0 \ 1 \end{pmatrix}$. What is $| \mathbf{v} |$?

A1: 0

A2: 1

A3: 5

A4: $\sqrt{10}$

A5: $\sqrt{26}$

Q3: What is not a property of the standard inner product $\langle \cdot, \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$?

A1: Positive definite

A2: Symmetric

A3: Linear in the second argument

A4: Linear in the first argument

A5: $\langle \mathbf{v}, \mathbf{v}\rangle = 0 ~ \Leftrightarrow ~ \mathbf{v}=0$

A6: $\langle \mathbf{v}, \mathbf{v}\rangle = 0 ~ \Leftrightarrow ~ \mathbf{v}\neq 0$