---
tag: 04C3BAHV03
kind: definition
name: Complementary subspaces
msc: [15A03]
defines: 0M83V17KHX
uses: [041JTZCK9Q]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_subspace
    locator: Definition
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$.
The subspaces $U$ and $W$ are *complementary* when $V = U \oplus W$, that is, when $U + W = V$ and $U \cap W = \{0\}$.
Then $W$ is a *complement* of $U$ in $V$.

## Remarks

A complement is not unique: in $\R^2$, each line through $0$ other than the $x$-axis is a complement of the $x$-axis.
If $V$ is finite-dimensional and $W$ is a complement of $U$, then $\dim W = \dim V - \dim U$.
A complement $W$ of $U$ is isomorphic to the [[quotient space|0BR3FY5Q8M]] $V / U$.
In an inner product space, the [[orthogonal complement|0XJXTBCB2T]] is one particular complement.

## Explanations

### L1 human

Two subspaces are complementary when together they fill the space and they share only the zero vector.
Each vector then splits in a unique way into a part in each subspace.
A line and a plane through the origin in space are complementary when the line does not lie in the plane.