Grassmann's formula

Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be finite-dimensional linear subspaces of $V$. Then $$\dim(U + W) + \dim(U \cap W) = \dim U + \dim W,$$ where $U + W$ is the sum of the subspaces and $\dim$ is the dimension.

Remarks

The proof extends a basis of $U \cap W$ to a basis of $U$ and to a basis of $W$. Together, these vectors form a basis of $U + W$. Another proof applies the rank–nullity theorem to the map $U \times W \to V$, $(u, w) \mapsto u - w$. Two planes through the origin in $\R^3$ with $U \ne W$ have $\dim(U + W) = 3$, so $\dim(U \cap W) = 1$.

Explanations

L1 human

The formula counts dimensions like a count of elements in a union of two sets. We add the two dimensions, but then the common part is counted twice, so we subtract it once. Two different planes through the origin in space always share at least a line.