Find out whether W is a subspace or not.

Question

Could you please walk me through the process of finding out whether

$w=\{(x,y,0,z^2)|x,y\in R,z\in Z\}$

is a subspace or not.

I got it in an exam and didn't know how to conquer the problem, also I couldn't find a similar problem almost anywhere.

Thanks.

Answer 1

$0 \in w$, so that's an alright start, but if we are taking this as a subset of $\mathbb{R}^4$, consider closure under scalar multiplication. In particular, the scalars come from $\mathbb{R}$, but the fourth coordinate of the points in this set can only be an integer.

There are also issues with closure under addition: given any $a, b \in \mathbb{Z}$, do we always have $a^2 + b^2 = c^2$ for some $c \in \mathbb{Z}$?

Answer 2

Hint

Clearly $(1,1,0,1) \in w$, but what about $2\cdot(1,1,0,1)=(2,2,0,2)$?