Question 1

Match the systems of linear equations in Column A with their number of solutions in column B and their geometric representation in Column C. Table matching systems of linear equations, number of solutions and 3D plane plots (Table M2W4G2) Consider the following two statements: **Statement 1:** If $V$ is defined as follows: $$V = \{(x,y) \mid x,y \in \mathbb{R}, x + y = 0\}$$ *Addition:* $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$; $(x_1, y_1), (x_2, y_2) \in V$ *Scalar multiplication:* $c(x, y) = (x, cy)$; $(x, y) \in V$, $c \in \mathbb{R}$ Then $V$ is a vector space. **Statement 2:** If $W = Span\{(1,0,1), (0,1,1), (2,-1,1), (2,-2,-1)\}$, then $dim(W) = 3$. Which of the following options is true? Choose the set of correct options.