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:
(d) The set of all vectors v = (x, y) such that |x| = |y|

Solution

Not a subspace

Consider two elements

(-3,3) and (-4,-4)

Both elements are in the set.

Now adding them gives

(-7,-1) which is not in the set

So the set is not closed under addition

Hence not a subspace