subspace of $\mathbb R^4$.

Question

$W=\{(a,b,c,d)|ab=cd\}.$

How do I prove that $W$ is subspace of $\mathbb R^4$?

I have $(0,0,0,0)\in W$, so $W$ not empty.

and $k\in\mathbb R$,$w\in W$ so $kw\in W$.

so I'm not sure that if $w_1,w_2\in W$ so $w_1+w_2\in W$ ?

Answer 1

It isn't, as it's not defined by linear relations.

Counter-example: set $\;w_1=(1,0,0,1)$, $w_2=(0,2,1,0)$. We have: $$w_1+w_2=(1,2,1,1)$$ and it doesn't satisfy the relation.

Answer 2

$(1,0,0,0) \in W$ because $1 \cdot 0 = 0 \cdot 0$.

Also

$(0, 1,0,0) \in W$ because $0 \cdot 1 = 0 \cdot 0$.

$(1,0,0,0)+(0,1,0,0)=(1,1,0,0)$ but $1\cdot 1 \ne 0 \cdot 0$.

Hence $W$ is not closed under addition.