Dimension of a subspace smaller than dimension of intersection Suppose I have a finite-dimensional vector space $V$, and $U_1, U_2$ are subspaces of $V$, such that $U_1\nsubseteq U_2$.
Is it possible that $\dim{U_1}\leq\dim{U_1 \cap U_2}$?

Fact: If $X$ is a subspace of the finite dimensional vector space $Y,$ and $\dim X = \dim Y,$ then $X=Y.$ Hence if $\dim U_1 \le \dim U_1\cap U_2,$ then $\dim U_1 = \dim U_1\cap U_2,$ hence $U_1\cap U_2$ equals $U_1.$ This implies $U_1 \subset U_2,$ contradiction. 
