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Suppose $U$ and $V$ are $2$ dimensional subspaces of $\mathbb{R}^4$. How do you determine the dimension of $U \cap V$? I know that

\begin{equation*} \text{dim}(U + V) = \text{dim}(U)+ \text{dim}(V) - \text{dim}(U \cap V). \end{equation*}

So it seems that $\text{dim}(U \cap V) = 0$.

  • 4
    There is not enough information to uniquely determine the answer. The dimension of the intersection can be 0, 1, or 2.2010-10-08
  • 0
    how can it be 1 or 2?2010-10-08
  • 3
    @Tom for 2, let U=V. For 1, consider U and V as planes which are intersecting in a line.2010-10-09

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