Second isomorphism theorem for vector spaces

Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$. Then the map $$U / (U \cap W) \to (U + W) / W, \qquad u + (U \cap W) \mapsto u + W,$$ is well defined, and it is a linear isomorphism. Here $U + W$ is the sum of the subspaces, and both sides are quotient spaces.

Remarks

In finite dimensions, it gives Grassmann's formula: $\dim U - \dim(U \cap W) = \dim(U + W) - \dim W$. The proof applies the first isomorphism theorem to the map $U \to (U + W)/W$, $u \mapsto u + W$, which is surjective with kernel $U \cap W$.

Explanations

L1 human

Adding $U$ to $W$ adds the same new directions as $U$ has beyond its common part with $W$. The theorem makes this precise with quotient spaces. It is the linear algebra form of a general algebraic theorem.