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:
(a) The set of all vectors v = (x, y) such that 2x + 3y = 0

Solution

1. Check for closure under addition

Let, (x,y) and (u,v) be in set

(x,y)+(u,v)=(x+u,y+v)

2(x+u)=2x+2u=3y+3v=3(y+v)

Hence,(x+u,y+v) is in the set so set is closed under addition

2. Check for closure under scalar multiplication

Let, c be a scalar and (x,y) in the set

c(x,y)=(cx,cy)

2(cx)=c*2x=c*3y=3*(cy)

So, set is closed under scalar multiplcation

HEnce set is a subspace of R2

Sketch

http://www.wolframalpha.com/input/?i=plot+2x%2B3y%3D0