Maths 2 Quiz 1: 11 July 2021, Set POD21QZBQPA (May 2021 term)

Match the systems of linear equations in Column A with their number of solutions in column B and their geometric representation in Column C.

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: $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, 0)$; $(x_1, y_1), (x_2, y_2) \in V$

Scalar multiplication: $c(x, y) = (cx, cy)$; $(x, y) \in V$, $c \in \mathbb{R}$

Then $V$ is a vector space.

Statement 2:

If $W = Span\{(1,0,1), (0,1,1), (2,-1,1), (1,-2,-1)\}$, then $dim(W) = 3$.

Which of the following options is true?

Choose the set of correct options.