---
tag: 0APN65D6Y6
kind: statement
role: theorem
name: Second isomorphism theorem for vector spaces
msc: [15A03, 16D10]
uses: [0SVR29SV1V, 0RMKABS2S5, 0V33SPH7RR]
sources:
  - url: https://en.wikipedia.org/wiki/Isomorphism_theorems
    locator: Theorem B (modules)
---

## 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|0JMTY8MF54]], and both sides are [[quotient spaces|0BR3FY5Q8M]].

## Remarks

In finite dimensions, it gives [[Grassmann's formula|06HTJNCMPA]]: $\dim U - \dim(U \cap W) = \dim(U + W) - \dim W$.
The proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] 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.