Linear and multilinear algebra; matrix theory Definition

Complementary subspaces

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 $V / U$ . In an inner product space, the orthogonal complement is one particular complement.

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Vector space definition
Linear subspace definition
Sum of subspaces definition
Direct sum of subspaces definition
Complementary subspaces definition

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Existence of a complementary subspace

Tag	`04C3BAHV03`
URI	`m2://04C3BAHV03`
Kind	definition
Writing	W1 stated
MSC	15A03
Defines	Complementary subspaces
Uses	Direct sum of subspaces

Complementary subspaces

Definition

Remarks

Reading path

Where this leads

Further reading

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