Problem-2

Let \(U\) and \(V\) be two three dimensional subspaces of \(\mathbb{R}^{5}.\) Show that there exists a non-zero vector \(v\in\mathbb{R}^{5}\) which lies in both \(U\) and \(V\).

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We have:

\[ \text{dim}(U\cap V)=\text{dim}(U)+\text{dim}(V)-\text{dim}(U+V) \]

The maximum value of \(\text{dim}(U+V)\) is \(5\), since \(U+V\) is a subspace of \(\mathbb{R}^{5}.\) Therefore, the minimum value of \(\text{dim}(U\cap V)\) is \(3+3-5=1\). Therefore, \(U\cap V\) is a non-trivial subspace of \(\mathbb{R}^{5}\). It follows that there exists some non-zero \(v\in\mathbb{R}^{5}\) that is in both \(U\) and \(V\).