Why this subset of $\mathbb{R}^3$ is not a subspace? Problem
Determine whether the indicated subset is a subspace of the given euclidean space:
$ \{[x,y,z]\ |\ x,y,z \in \mathbb{R} $ and $z=3x+2\}$ in $\mathbb{R}^{3}$
Solution
By definition, in order for a subset to be a subspace 3 conditions must be occur:


*

*To pass by the origin To contain the origin.

*To be closed under addition. 

*To be closed under scalar multiplication.


So I try to solve the exercise by this way:
$1.$ The origin $(0,0,0) \in \mathbb{W} $
$2.$ Let $\vec u$ and $\vec v \in \mathbb{W} $. We have
$$ 
\begin{cases}
3u_1 + 2 - u_3 = 0 \\
3v_1 + 2 - v_3 = 0 \\
\end{cases}
$$
The sum is $ 6(u_1 + v_1) + 4 - (u_3+v_3) = 0 $ (which $\in \mathbb{W} $)
$3.$ Let $\vec u$ $ \in \mathbb{W} $ and  $\ r$ $ \in \mathbb{R} $. We have
$r(3u_1) + r(2) - r(u_3) = 0 \\$
Which, also, $ \in \mathbb{W} $
So, why is the book's answer: It isn't a subspace? 
 A: *

*The origin is not in the set. In order to be in the set, you need $z=3x+2$. For $(0,0,0)$, $x=z=0$, but $0\neq 3(0)+2$.

*The set is not closed under sums. In order for 
$$(u_1,u_2,u_3)+(v_1,v_2,v_3)=(u_1+v_1,u_2+v_2,u_3+v_3)$$ to be in the set, you need $u_3+v_3 = u_1+v_1 + 2$. Your sum of equations proves nothing, and saying that the equation (which is false) "is in $\mathbb{W}$" is false; equations (or values) are not in $\mathbb{W}$, only vectors can be in $\mathbb{W}$. 
For an explicit example, note that $(0,0,2)$ and $(1,0,5)$ are both in $\mathbb{W}$. The sum is $(1,0,7)$, but $(1,0,7)$ is not in $\mathbb{W}$, because $7\neq 3(1)+2$. 

*The set is not closed under scalar multiplication. In order for
$$r(u_1,u_2,u_3) = (ru_1,ru_2,ru_3)$$
to be in $\mathbb{W}$, you need $ru_3 = 3(ru_1)+2$. From the assumption that $u_3=3u_1+2$ you cannot conclude that $ru_3 = 3(ru_1)+2$. In fact, this is false for any $r\neq 1$. For an explicit example, note that $(0,0,2)$ is in $\mathbb{W}$, but $2(0,0,2) = (0,0,4)$ is not, because $4\neq 3(0)+2$. 
So the reason it is not a subspace is that it fails every single one of the three requirements. 
