Linear algebra questions

Question

$M$ and $N$ subspaces of $V$. Give examples of $M$ and $N$ such that $M\cup N$ and $M\setminus N$ are not subspaces.

Answer 1

Hint

• Take any subspaces $M$ and $N$ of $V$ such that $M\not\subset N$ and $N\not\subset M$ and then $M\cup N$ isn't a subspace of $V$(why?)
• Any subspace contains the zero vector.

Answer 2

Take $V = (R^2,+)$, and $M = \{(x,0): x \in \mathbb{R}\}$, and $N = \{(0,y): y \in \mathbb{R}\}$. To see that $M\cup N$ is not a subspace, take $(1,0) \in M, (0,1) \in N$, then $(1,0)+(0,1) = (1,1) \not \in M\cup N$, and also $M\setminus N$ is not a subspace since $(0,0) \not \in M\setminus N$