Suppose UU and WW are both 4-dimensional subspaces of C6\mathbf C^6. Prove that there exist two vectors in U∩WU \cap W such that neither of these vectors is a scalar multiple of the other.

By 2.43

dim⁡(U∩W)=dim⁡U+dim⁡W−dim⁡(U+W)=8−dim⁡(U+W)≥2\begin{aligned} \dim(U \cap W) &= \dim U + \dim W - \dim(U+W) \\ &= 8 - \dim(U + W) \\ &\ge 2 \end{aligned}

Therefor we can find two independent vectors v1,v2∈U∩Wv_1,v_2 \in U \cap W which clearly won't be scalar multiples of eachother.