Question 1

Matching table of linear systems with 3D plots of planes as geometric representations Figure from the original question paper Consider the set $V$ together with the operations addition and scalar multiplication defined as follows: $$V = \{(x,y,z) \mid x,y,z \in \mathbb{R}\}$$ *Addition:* $(x_1, y_1, z_1) + (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2)$; $(x_1, y_1, z_1), (x_2, y_2, z_2) \in V$ *Scalar multiplication:* $$c(x,y,z) = \begin{cases} (0,0,0) & c = 0 \\ (cx, cy, \frac{z}{c}) & c \neq 0 \end{cases} \quad (x,y,z) \in V, \ c \in \mathbb{R}$$ 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 $v$ in $V$, there exists an element $v'$, such that $v' + v = v + v' = 0$. 3. For each $v \in V$, $1v = v$. 4. For each $v \in V$ and for each pair $a, b \in \mathbb{R}$, $(a+b)v = av + bv$. 5. For each $a \in \mathbb{R}$ and for each pair $v_1, v_2 \in V$, $a(v_1 + v_2) = av_1 + av_2$. 6. For each $v \in V$ and for each pair $a, b \in \mathbb{R}$, $(ab)v = a(bv)$. Choose the correct statements from the above.