SIMON-MARAIS 2023 B3

Let $n$ be a positive integer. Let $A, B$ and $C$ be three $n$-dimensional subspaces of $\mathbb{R}^{2n}$ with the property that $A \cap B = B \cap C = C \cap A = \{0\}$. Prove that there exist bases $\{a_1, a_2, \ldots, a_n\}$ of $A$, $\{b_1, b_2, \ldots, b_n\}$ of $B$ and $\{c_1, c_2, \ldots, c_n\}$ of $C$ with the property that for each $i \in \{1, 2, \ldots, n\}$, the vectors $a_i, b_i$ and $c_i$ are linearly dependent.