Question

How do I show that a set is closed under vector addition?

Answer 1

Suppose you have two vectors say u and v, then their sum i.e (u + v) must also belong to the same set for all u and v in the given set. Then, the given set is said to be closed under vector addition.
Ex: Let the set be R (set of real numbers),
u = (-3, 8.4, 5) and v = (1, 0, 1) be 2 vectors from the set R.
u + v = (-2, 8.4, 6) , Here it can be observed that the result also belongs to R.
This is true for all the vectors in R, so we can say R is closed under vector addition.