Intersection of Subspaces

Theorem

• If $W_1,W_2$ are Subspace of $V$
• Then, $W_1\cap W_2$ is also a Subspace of $V$

Proof

1. First check if $W_1\cap W_2$ is non-empty
   1. We know $\overrightarrow{0}\in W_1$ and $\overrightarrow{0}\in W_2$ because $W_1,W_2$ are subspaces of $V$
   2. Therefore, $\overrightarrow{0}\in W_1\cap W_2$ so $W_1\cap W_2$ is non-empty
2. Apply Subspace Test
   1. Pick $\overrightarrow{x},\overrightarrow{y}\in W_1\cap W_2$
   2. Pick $c\in F$
   3. We know $\overrightarrow{x},\overrightarrow{y}\in W_1$, so $c\overrightarrow{x}+\overrightarrow{y}\in W_2$
   4. We know $\overrightarrow{x},\overrightarrow{y}\in W_2$, so $c\overrightarrow{x}+\overrightarrow{y}\in W_1$
   5. Therefore, $c\overrightarrow{x}+\overrightarrow{y}\in W_1\cap W_2$
   6. Thus, it follows that $W_1\cap W_2$ is a subspace