Let E = xy > 0. Determine whether E is a subspace of R2 . X3

Question

Let E = (x, y) e R2. Determine whether E is a subspace of R2 . X3

Answer 1

E is not a subspace of
`\mathbb{R}^2`

Explanation:

E is not a subspace of
`\mathbb{R}^2`

In order to see this, we must find two points (a,b), (c,d) in E such that (a,b) + (c,d) is not in E.

Consider

(a,b) = (1,1)

(c,d) = (-1,-1)

It is easy to see that both (a,b) and (c,d) are in E since 1*1>0 and (1-)*(-1)>0.

But (a,b) + (c,d) = (1-1, 1-1) = (0,0)

and (0,0) is not in E.

By the way, it can be proved that in any vector space all sub spaces must have the vector zero.