Q1: Which claim is correct for vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$?

A1: $\mathbf{u}^T \mathbf{v} = \langle \mathbf{u}, \mathbf{v} \rangle$

A2: $\mathbf{u} \mathbf{v}^T = \langle \mathbf{u}, \mathbf{v} \rangle$

A3: $\mathbf{u} \mathbf{v} = \langle \mathbf{u}, \mathbf{v} \rangle$

A4: $\mathbf{u}^T \mathbf{v}^T = \langle \mathbf{u}, \mathbf{v} \rangle$

Q2: Let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^2$ given by $\mathbf{u} = \binom{1}{2}$ and $\mathbf{v} = \binom{-2}{3}$. What is $\mathbf{u}^T \mathbf{v}$?

A1: $4$

A2: $0$

A3: $1$

A4: $3$

A5: $2$

Q3: For a matrix $A$ with rows $\alpha_1^T, \ldots, \alpha_m^T$, we calculate the matrix-vector product $A \mathbf{x}$. What do we get?

A1: $$\begin{pmatrix} \alpha_1^T \mathbf{x} \ \alpha_2^T \mathbf{x} \ \ldots \ \alpha_m^T \mathbf{x}\end{pmatrix} $$

A2: $$\begin{pmatrix} \alpha_m^T \mathbf{x} \ \alpha_m^T \mathbf{x} \ \ldots \ \alpha_m^T \mathbf{x}\end{pmatrix} $$

A3: $$\begin{pmatrix} \alpha_m^T \mathbf{x}^T \ \alpha_m^T \mathbf{x}^T \ \ldots \ \alpha_m^T \mathbf{x}^T \end{pmatrix} $$

A4: $$\begin{pmatrix} \alpha_1^T \mathbf{x} \ \alpha_1^T \mathbf{x} \ \ldots \ \alpha_1^T \mathbf{x}\end{pmatrix} $$