Question-57

Let \(\displaystyle S_{1}\) and \(\displaystyle S_{2}\) be two subspaces of \(\displaystyle \mathbb{R}^{10}\) with dimensions \(\displaystyle 7\) and \(\displaystyle 3\) respectively. Let \(\displaystyle d\) be the dimension of \(\displaystyle S_{1} \cap S_{2}\).

• The maximum possible value of \(\displaystyle d\) is \(\displaystyle 7\).
• The maximum possible value of \(\displaystyle d\) is \(\displaystyle 3\).
• The minimum possible value of \(\displaystyle d\) is \(\displaystyle 1\)
• The minimum possible value of \(\displaystyle d\) is \(\displaystyle 0\).

Answer

• The maximum possible value of \(\displaystyle d\) is \(\displaystyle 7\).
• The maximum possible value of \(\displaystyle d\) is \(\displaystyle 3\).
• The minimum possible value of \(\displaystyle d\) is \(\displaystyle 1\)
• The minimum possible value of \(\displaystyle d\) is \(\displaystyle 0\).

Solution

• Since the sum of the dimensions of the two subspaces is equal to the dimension of the parent subspace, it is possible for the two subspaces to have a trivial intersection. This suggests that the minimum possible value of \(\displaystyle d\) is \(\displaystyle 0\).

• If the smaller subspace is contained in the bigger subspace, that would give us the maximum possible value for \(\displaystyle d\), which is \(\displaystyle 3\).