# Is the set of all vectors in R2 of the form (a, b) where b = a closed under addition?

Apr 1, 2017

Yes. For Proof , refer to The Explanation.

#### Explanation:

Let $V = \left\{\left(a , b\right) : a = b \in \mathbb{R}\right\} = \left\{\left(a , a\right) : a \in \mathbb{R}\right\} \subset {\mathbb{R}}^{2.}$

Let $\vec{x} = \left(a , a\right) \mathmr{and} \vec{y} = \left(b , b\right)$ be arbitrary vectors of $V , w h e r e , a , b \in \mathbb{R} .$

By the Defn. of Addition of Vectors,

$\vec{x} + \vec{y} = \left(a , a\right) + \left(b , b\right) = \left(a + b , a + b\right) = \left(c , c\right) , \text{ say, where, } c = a + b \in \mathbb{R} .$

Clearly, $\vec{x} + \vec{y} \in V .$

This shows that $V$ is closed under vector addition.

Enjoy Maths.!