# Thread: Proving intersection is a subspace

1. ## Proving intersection is a subspace

Let $W_1$ and $W_2$ be two subspaces of a vector space $V$ over the field $\mathbb{F}$. Prove that the intersection of $W_1$ and $W_2$ is a subspace of $V$.

Can someone check my working for this question

Let $x\in W_1, y\in W_1$

Since $W_1$ is a subspace, it is closed under addition and so $x+y\in W_1$

Similarly, let $x\in W_2, y\in W_2$

Since $W_2$ is a subspace, it is closed under addition and so $x+y\in W_2$

$\Rightarrow x+y\in W_1\cap W_2\Rightarrow$ Closure under addition

Let $\lambda\in\mathbb{R}, x\in W_1, W_2$

$\lambda x\in W_1$ since $W_1$ is a subspace

$\lambda x\in W_2$ since $W_2$ is a subspace

Clearly, $\lambda x\in W_1\cap W_2\Rightarrow$ Closure under scalar multiplication

$\Therefore W_1 \cap W_2$ is a subspace of $V$

What about the condition for non-empty? This condition always confuses me. Why is it that the zero vector has to be included for it to be non-empty?

2. Originally Posted by acevipa
Let $W_1$ and $W_2$ be two subspaces of a vector space $V$ over the field $\mathbb{F}$. Prove that the intersection of $W_1$ and $W_2$ is a subspace of $V$.

Can someone check my working for this question

Let $x\in W_1, y\in W_1$

Since $W_1$ is a subspace, it is closed under addition and so $x+y\in W_1$

Similarly, let $x\in W_2, y\in W_2$

Since $W_2$ is a subspace, it is closed under addition and so $x+y\in W_2$

$\Rightarrow x+y\in W_1\cap W_2\Rightarrow$ Closure under addition

Let $\lambda\in\mathbb{R}, x\in W_1, W_2$

$\lambda x\in W_1$ since $W_1$ is a subspace

$\lambda x\in W_2$ since $W_2$ is a subspace

Clearly, $\lambda x\in W_1\cap W_2\Rightarrow$ Closure under scalar multiplication

$\Therefore W_1 \cap W_2$ is a subspace of $V$

What about the condition for non-empty? This condition always confuses me. Why is it that the zero vector has to be included for it to be non-empty?
well, since $\displaystyle W_1$ and $\displaystyle W_2$ are subspaces, they each contain 0. so 0 will be in their intersection, and hence, the intersection is non-empty.

containing zero is important for vector spaces and are hence important for subspaces. so the subspaces must have 0 in them. it is not to say that they have to have 0 to be non-empty per sae, but it is usually easy to verify that the zero vector is in there. so you always check that. and if 0 is not in there, then you're in trouble. you wouldn't have a vector space at all. just some other kind of set.

3. Thanks a lot. I understand that well now.