Some Usefull Theorems

If \(U, V\) are two linear subspaces, then \(\dim(U + V) = \dim(U) + \dim(V) - \dim(U \bigcap V)\).

This can be proved by considering the basis of \(U \bigcap V\) first and then extends the basis to \(U\) and \(V\) respectively.

The intersection of affine subspace \(U\) of dimension \(m < n\) and hyperplane \(H\) (affine subspace of dimension \(n - 1\)) has three cases:

1. They don't intersect.
2. \(H\) contains \(U\). The dimension of intersection is \(m\).
3. The dimension of intersection is \(m - 1\).

This can be proved in this way:

Suppose \(U\) and \(H\) intersects, then we can assume they are not affine but linear instead (by shifting the center point to the intersection set).

Then \(\dim (U + H) = n = \dim(U) + \dim(H) - \dim(U \bigcap H) = m + n - 1 - \dim(U \bigcap H)\).

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