Q30Multiple correct2 Marks7 Aug 2022

An inner product on a vector space V is a function <·, ·> : V x V -> R satisfying the following conditions: Condition 1: $\langle v, v \rangle$ > 0 for all v in V \ {0}; $\langle v, v \rangle$ = 0 if and only if v = 0. Condition 2: $\langle v1 + v2, w \rangle$ = $\langle v1, w \rangle$ + $\langle v2, w \rangle$ . Condition 3: $\langle v1, v2 \rangle$ = $\langle v2, v1 \rangle$ . Condition 4: $\langle cv1, v2 \rangle$ = c $\langle v1, v2 \rangle$ . Let V = $\mathbb{R}^2$ and consider the function defined as: <·, ·> : V x V -> R, <(x1, x2), (y1, y2)> = x1 y1 - 4x2 y2. Which of the following are satisfied by the above function?

A

Condition 1 is satisfied.

B

Condition 2 is satisfied.

C

Condition 3 is satisfied.

D

Condition 4 is satisfied.