Q1: Let $U \subseteq \mathbb{R}^n$ be a subspace. Which statement is not correct?

A1: $\mathbf{0} \in U$.

A2: For $\mathbf{u}, \mathbf{v} \in U$, we have $\mathbf{u} - 2 \mathbf{v} \in U$.

A3: For $\mathbf{u}, \mathbf{v} \in U$, we have $5 \mathbf{u} - 8 \mathbf{v} \in U$.

A4: For $\mathbf{u} \in U$, we have $3\mathbf{u} \notin U$.

Q2: Which of the following subsets in $\mathbb{R}^2$ is not a subspace of $\mathbb{R}^2$?

A1: ${ \mathbf{v} \in \mathbb{R}^2 \mid \mathbf{v} \neq \mathbf{0} }$

A2: ${ \mathbf{0} }$

A3: ${ \mathbf{v} \in \mathbb{R}^2 \mid \mathbf{v} = \mathbf{0} }$

A4: $ \mathbb{R}^2 $

A5: $\left{ \binom{v_1}{v_2} \in \mathbb{R}^2 \mid v_1 + v_2 = 0 \right}$

Q3: Let $U,V \subseteq \mathbb{R}^n$ be two subspaces. Is the intersection $U \cap V$ a subspace as well?

A1: Yes!

A2: No!