Q1: Is the set $U$ given by $$ \left{ \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} ~\bigg|~ x_1 = - x_2 \text{, } x_3 = 2 x_2 \right}$$ a subspace of $\mathbb{R}^3$?

A1: Yes!

A2: No!

A3: One needs more information.

Q2: Is the set $U$ given by $$\left{ \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix}~\bigg|~ x_1 = - x_2^3 \text{, } x_3 = 0 \right}$$ a subspace of $\mathbb{R}^3$?

A1: No!

A2: Yes!

A3: One needs more information.

Q3: Is the set $U$ given by $$\left{ \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix}~\bigg|~ \begin{matrix} \exp(x_1) - 1 = x_2 \text{, }\ x_2^2 + x_3^2 = 0 \end{matrix} \right}$$ a subspace of $\mathbb{R}^3$?

A1: Yes!

A2: No!

A3: One needs more information.