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:
(b) The set of all vectors v = (x, y) such that x + y = 1

Solution

b)

Not a subspace

Consider a point int he set: (0,1)

0+1=1

For set to be subspace it must be closed under scalar multiplication

2*(0,1)=(0,2)

0+2=0

so the set is not closed under scalar multiplication and hence nto a subspace