Which of the following subsets of $\mathbb{R}^3$ are subspaces? Which of the following subsets of $\mathbb{R}^3$ are subspaces? Explain the answer.
$(a)\ \{(x,y,z) \mid x = 1\}$
$(b)\ \{(x,y,z)\mid z=3x−2y\}$
$(c)\ \{(x,y,z) \mid xy = 0\}$ 
$(d)\ \{(x,y,z)\mid z=x−2y, z=2x−y\}$
 A: Subspace Test
Let $V$ be a vector space and let $W \subset V$ where $W \ne \varnothing$. $W$ is a subspace of $V$ if


*

*For $\mathbf x_1$, $\mathbf x_2 \in W$, $\mathbf x_1 + \mathbf x_2 \in W$

*For $\mathbf x_1 \in W$ and $k \in R$, $k\mathbf x_1 \in W$


i.e. $W$ must be non-empty and closed under addition and scalar multiplication.

$$(b) W =\lbrace (x,y,z) | z =3x -2y \rbrace$$
$W$ is clearly non-empty; 
$$0 = 3(0) - 2(0) \implies (0,0,0) \in W$$
Let $\mathbf x_1 = (x_1, y_1, 3x_1 - 2y_1)$ and $\mathbf x_2 = (x_2, y_2, 3x_2 - 2y_2)$
$$\mathbf x_ 1 + \mathbf x_2 = (x_1 + x_2, y_1 + y_2, 3x_1-2y_1 + 3x_2-2y_2)$$ 
$$= (x_1 + x_2, y_1 + y_2, 3(x_1+x_2) - 2(y_1 +y_2)) \in W$$
For $k \in R$
$$k\mathbf x_1 =  k \left(x_1, y_1, 3x_1 - 2y_1 \right)$$
$$= (kx_1, ky_1 3(kx_1) - 2(ky_1)) \in W$$
By Subspace Test, $W$ is a subspace of $R^3$
A: For $a)$, you are correct. Because $\vec{0}$ is not in the set, regardless of what value $y$ or $z$ are, it fails the Subspace Test, and is therefore not a subspace of $\mathbb{R}^3$.
