Suppose UU and WW are both five-dimensional subspaces of R9\mathbf R^9. Prove U∩W≠{0}U \cap W \ne \{0\}

By 2.43

dim⁡U∩W=dim⁡U+dim⁡W−dim⁡(U+W)=10−dim⁡(U+W)≥1\begin{aligned} \dim U \cap W &= \dim U + \dim W - \dim(U+W) \\ &= 10 - \dim(U + W) \\ &\ge 1 \end{aligned}

Since dim⁡U∩W≥1\dim U\cap W \ge 1 we must have U∩W≠{0}U \cap W \ne \{0\}.