Let H and K be subspaces of a vector space $V$. The intersection of $H$ and $K$, , is the set of $v$ in $V$ that belong to both $H$ and $K$. Show that the intersection of $H$ and $K$ is a subspace of $V$. Give an example in $R^2$ to show that the union of two subspaces is not, in general, a subspace.
I know that in order to prove that a set is a subspace of a vector space, we need to prove that it contains the zero vector, closed under multiplication and addition. I know that the zero vector exists in the intersection from the graph above.
And it seems to me that the intersection of $H$ and $K$ is along the z-axis (i think…) so i am assuming that i can take any two vectors along the z-axis (0,0,1), (0,0,2)… and add it together to prove it is closed under addition, and the same logic applies to using any scalar and multiplying with any vector along the z-axis to prove it is closed under multiplication
Is this the right approach? or did i make a mistake somewhere. Any help will be appreciated.
Best Answer
You should prove things in general, i.e. using the properties that $H$ and $K$ are subspaces to prove that $H \cap K$ is a subspace.
$0 \in H$ and $0 \in K$, hence $0 \in H \cap K$.
Now let $u \in H \cap K$ and $v \in H \cap K$, justfity that $u +v \in H \cap K$.
Similarly verify for scalar multiplication.
As for counter example for the union, just have to find one in $\mathbb{R}^2$, consider $2$ different straight lines that passes through the origin, what can you say about their union? which property fails?