Question

Identify a subspace of 2 out of four given options:
1. A line in 2 passing through the points (0, 1) and (-1, 0).
2. A line in 2 passing through the points (0-1) and (-1, 0).
3. A line in 2 passing through the points (1, 3) and (-1, -3).
4. A line in 2 passing through the points (0, -1) and (1, 0).

Answer 1

Correct Answer - Option 3 : A line in 2 passing through the points (1, 3) and (-1, -3).

Concept:

A subset S of a vector space V is a subspace of V if and only if S is nonempty and closed under linear operations, i.e., 

x, y ∈ S ⇒ x + y ∈ S,      ----(closed under addition)

x ∈ S ⇒ rx ∈ S      for all r ∈ R      ----(closed under multiplication)

Calculation:

Let's check from the options

(A) (0, 1) and (-1, 0)

Here, (0 + (-1)), (1 + 0) = (-1, 1)

⇒ not closed under addition. 

(B) (0-1) and (-1, 0)

Here, (0 + (-1)), (-1 + 0) = (-1, -1)

⇒ not closed under addition.

(C) (1, 3) and (-1, -3)

Here, (1 + (-1)), (3 + (-3)) = (0, 0)

⇒ Closed under addition.

(D) (0, -1) and (1, 0)

Here, (0 + 1), (-1 + 0) = (-1, 1)

⇒ not closed under addition.

Hence, a line in 2 passing through the points (1, 3) and (-1, -3) is a subspace for 2.