Question

Let W = {(x, y, z): y² = x + z}, Is W a subspace of R³

Answer 1

$W$ consists of all vectors $(x,y,z)$ satisfying: $y^2=x+z$. Suppose that $(x_1,y_1,z_1)$ and $(x_2,y _2,z_2)$ belong to $W$. It means that $y_1^2=x_1+z_1$ and $y_2^2=x_2+z_2$. But, it does not mean that $(x_1+x_2,y_1+y_2,z_1+z_2)$ belongs to $W$. I.e., it may not satisfy $(y_1+y_2)^2=(x_1+x_2)+(z_1+z_2)$. For example, we can choose $(1,2,3)$ and $(0,1,1)$. It is clear that both vectors belong to $W$. But $(1,3,4)$ does not belong to $W$, since $3^2\neq1+4$. The latter means that one of subspace properties is not satisfied for $W$ . Therefore, it is not a subspace of $\mathbb{R}^3$ .