Let $U,V$ be two planes in $\mathbb R^3$ passing the origin. Show that their intersection $=:W$ (also a subspace) is at least one dimensional.
Hint: We have to show that $W$ contains a line, which is the case if it contains at leas one non-zero vector $ = \vec v$.
One possibility is as follows: Let $\vec a_1,\vec a_2$ be a basis in $U$ and $\vec b_1, \vec b_2$ be a basis in $V$. Show that we can find coefficients $\lambda_1, \lambda_2, \mu_1, \mu_2,$ $\vec\lambda\neq\vec 0\neq\vec\mu$, such that $\lambda_1 + \vec a_1 +\lambda_2 \vec a_2 = \vec v = \mu_1\vec b_1 + \mu_2 \vec b_2$ in particular $$\lambda_1 + \vec a_1 +\lambda_2 \vec a_2 - \mu_1\vec b_1 - \mu_2 \vec b_2 = \vec 0.$$
What can you say about the four vectors in three dimensions? ... Finish by explaining why we can fint nontrivial $\vec\lambda$ and $\vec\mu$.
What may I say about the four vectors in 3-D? Why exactly could we find nontrivial $\lambda$ and $\mu$?