Question 1

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? Figure from the original question paper