Question

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

Answer 1

Yes. For Proof , refer to The Explanation.

Explanation:

Let `V={(a,b) : a=b in RR} ={(a,a) : a in RR} sub RR^2.`

Let `vecx=(a,a) and vecy=(b,b)` be arbitrary vectors of `V, where, a,b in RR.`

By the Defn. of Addition of Vectors,

`vecx+vecy=(a,a)+(b,b)=(a+b,a+b)=(c,c)," say, where, "c=a+b in RR.`

Clearly, `vecx+vecy in V.`

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

Enjoy Maths.!