Question 1

Choose the set of correct options. If addition and scalar multiplication on $V = \mathbb{R}^2$ is defined as follows: *Addition:* $(x_1, y_1) + (x_2, y_2) = (0, 0)$; $(x_1, y_1), (x_2, y_2) \in V$ *Scalar multiplication:* $c(x, y) = (0, 0)$; $(x, y) \in V$, $c \in \mathbb{R}$ Consider the following statements. 1. There exists an element $0$ (called the zero vector of $V$) in $V$ such that $0 + v = v, \ \forall\, v \in V$. 2. For each vector of $v \in V$ and for each pair $a, b \in \mathbb{R}, (a + b)v = av + bv$. 3. For each vector of $a \in \mathbb{R}$ and for each pair $v_1, v_2 \in V, a(v_1 + v_2) = av_1 + av_2$. 4. For each vector of $v \in V$ and for each pair $a, b \in \mathbb{R}, (ab)v = a(bv)$. Which of the above statements is not true with respect to the addition and scalar multiplication on $V = \mathbb{R}^2$ defined above? (Enter the serial number of the statement which is not true. If statement 2 is incorrect, then enter 2 as your answer.) Consider the following two statements: **P:** $V = \mathbb{R}^2$, with the operations: **Addition:** $$(x_1, y_1) + (x_2, y_2) = (x_1x_2, y_1y_2); \; (x_1, y_1), (x_2, y_2) \in V$$ and **Scalar multiplication:** $$c(x, y) = (cx, cy); \; (x, y) \in V, \; c \in \mathbb{R}$$ is a vector space. **Q:** Let $V$ be a vector space. If $u, v, w \in V$ are such that $au + bv + cw = 0$ for some scalars $a, b, c \in \mathbb{R}$ and $ac \neq 0$, then $\text{span}\{u, v\} = \text{span}\{v, w\}$. Consider the following statements: - **Statement 1:** P is true, but Q is false. - **Statement 2:** Q is true, but P is false. - **Statement 3:** Both P and Q are true. - **Statement 4:** Both P and Q are false. Which one of the above statements is correct? (e.g. if Statement 1 is correct, then enter 1 as your answer).