Question

Determine whether or not the set W = {(x, y, z) ϵ Rᶟ xy = z} is a subgroup of the group

Rᶟ under coordinatewise addition.

Answer 1

Given W={(x,y,z):xy=z} ...(1)

Recall : A subset is said to be subspace of a vector space if

1) x,y$\in$ W $\implies$ x+y$\isin$ W

2) if x$\isin$ W and a$\isin$ F $\implies$ax$\in$ W

Now let's check these property for given W

Let x=(x₁,y₁,z₁) and y=(x₂,y₂,z₂) $\isin$ W

Hence , x₁y₁=z₁ and x ₂y₂=z₂

But (x+y)=(x₁+x₂ ,y₁+y₂,z₁+z₂)

So, (x₁+x₂)(y₁+y₂)=(z₁+z₂) is NOT equal to x₁y₁+x₂y₂=(z₁+z₂)

Hence ,(x+y) does not belong to W

Hence W is not a subspace.