This problem is asking us to evaluate a set $V = \{(1, -y) \mid y \in \mathbb{R}\} \subseteq \mathbb{R}^2$, given usual vector addition and scalar multiplication, and determine which statements are true based on the axioms of a vector space.

Step-by-step breakdown of a vector space:

To determine whether $V$ satisfies the vector space axioms, we must check:

1. Closure under addition: If $u, v \in V$, then $u + v \in V$.
2. Closure under scalar multiplication: For any scalar $c \in \mathbb{R}$ and any vector $v \in V$, $c \cdot v \in V$.
3. Zero vector: There must exist a vector $0 \in V$ such that $v + 0 = v$ for all $v \in V$.
4. Additive inverses: For each $v \in V$, there must exist a vector $-v \in V$ such that $v + (-v) = 0$.

Checking the options:

• Option 1: $(1,2) \in V$.
  By the definition of $V$, the first component must always be 1, and the second component must be $-y$, where $y \in \mathbb{R}$. For $(1,2)$, there is no real number $y$ such that $-y = 2$. Hence, $(1,2) \notin V$, so this is false.

• Option 2: $(2,3) \in V$.
  Here, the first component is 2, but in $V$, the first component must always be 1. Therefore, $(2,3) \notin V$, and this is false.

• Option 3: $V = \mathbb{R}^2$.
  $V$ is not equal to $\mathbb{R}^2$ because $V$ contains only vectors of the form $(1, -y)$, meaning the first component is always 1. This restricts $V$ to a subset of $\mathbb{R}^2$, so this is false.

• Option 4: $v_1 = (1, -5)$ and $v_2 = (1, 7) \in V$, and $v_1 + v_2 \in V$.
  Adding $v_1$ and $v_2$, we get:
  $v_1 + v_2 = (1, -5) + (1, 7) = (2, 2)$ Since the first component is 2, $(2, 2) \notin V$ because $V$ requires the first component to be 1. Thus, this is false.

• Option 5: $v = (1, 9) \in V$, $c \in \mathbb{R}$, and $c v \in V$ implies $c = 1$.
  For $v = (1,9)$, scalar multiplication by $c$ gives: $c \cdot (1, 9) = (c, c \cdot 9)$ For this to be in $V$, the first component must be 1. Therefore, $c = 1$, and this is true.

Answer:

The correct answer is Option 5.

Would you like more details or have any questions?

Here are 5 related questions to expand on the topic:

1. How do we check closure under vector addition and scalar multiplication?
2. Why is the zero vector important in vector spaces?
3. What happens if the first component of vectors in $V$ was allowed to vary?
4. Can you provide an example of a vector space that differs from $\mathbb{R}^2$?
5. How can we verify if a subset is a subspace of a vector space?

Tip: When checking if a set forms a vector space, always carefully verify if the operations of addition and scalar multiplication remain within the set.