Determine whether the following subsets of R2 are subspaces

Determine whether the following subsets of R² are subspaces (and state at least
one condition fails if not). Sketch the set:
(c) The set of all vectors v = (x, y) such that xy=0

Solution

False

Consider the two elements in this set

(0,1) and (1,0)

(0,1)+(1,0)=(1,1)

1*1=1

Hence , (1,1) is not in the set and henc set is not closed under addition so not a subspace