So I'm doing a review test and I have this problem:

Prove in full detail, with the standard operations in R2, that the set

```
{(x,2x): x is a real number}

```

is a vector space.

Attempt:

Given:

$(x\_1, 2x\_1) \in \mathbb{R}^2$

and

$(x\_2, 2x\_2) \in \mathbb{R}^2$

Addition:

$(x\_1, 2x\_1) + (x\_2, 2x\_2) = (x\_1 + x\_2, 2x\_1 + 2x\_2) \in \mathbb{R}^2$

$ = (x\_1 + x\_2, 2(x\_1 + x\_2)) \in \mathbb{R}^2$

$ ≃ (x, 2x) \in \mathbb{R}^2$

Thus the set is closed under addition

Scalar multiplication:

$c(x\_1, 2x\_1) = (cx\_1, 2(cx\_1)) \in \mathbb{R}^2$

$ ≃ (x, 2x) \in \mathbb{R}^2$

Thus the set is closed under scalar multiplication

---

Are these operations enough to prove that the set is a vector space? Or do I have to go through each of the following (or in other words do I have to to the same thing for each property in the definition):

![enter image description here](https://i.stack.imgur.com/PDcix.png)