Mathematics for Data Science II Quiz 2: 10 July 2022 (May 2022 term)

Choose the correct options.

An inner product on a vector space $V$ is a function $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ satisfying the following conditions:

Condition 1: $\langle v, v \rangle > 0$ for all $v \in V \setminus \{0\}$; $\langle v, v \rangle = 0$ if and only if $v = 0$.

Condition 2: $\langle v_1 + v_2, v_3 \rangle = \langle v_1, v_3 \rangle + \langle v_2, v_3 \rangle$.

Condition 3: $\langle v_1, v_2 \rangle = \langle v_2, v_1 \rangle$.

Condition 4: $\langle cv_1, v_2 \rangle = c\langle v_1, v_2 \rangle$

Let $V = \mathbb{R}^2$ and consider the function defined as:

$\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$

$\langle (x_1, x_2), (y_1, y_2) \rangle = x_1y_1 - x_1y_2 - x_2y_1 + x_2y_2.$

Which of the following are satisfied by the above function?