Linear and multilinear algebra; matrix theory Definition

Direct sum of subspaces

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $U_1, \dots, U_k$ be subspaces of $V$ . The space $V$ is the direct sum of $U_1, \dots, U_k$ , written $V = U_1 \oplus \cdots \oplus U_k$ , when each $v \in V$ can be written as $v = u_1 + \cdots + u_k, \qquad u_i \in U_i,$ in exactly one way. Equivalently, $V = U_1 + \cdots + U_k$ , and $u_1 + \cdots + u_k = 0$ with $u_i \in U_i$ implies $u_1 = \cdots = u_k = 0$ .

Remarks

For two subspaces, $V = U \oplus W$ if and only if $V = U + W$ and $U \cap W = \{0\}$ . For three or more subspaces, pairwise zero intersections are not enough: three different lines through $0$ in $\R^2$ are not a direct sum. If $V = U_1 \oplus \cdots \oplus U_k$ and $V$ is finite-dimensional, then $\dim V = \dim U_1 + \cdots + \dim U_k$ .

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

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Tag	`041JTZCK9Q`
URI	`m2://041JTZCK9Q`
Kind	definition
Writing	W0 stub
MSC	15A03
Defines	Direct sum of subspaces
Uses	Sum of subspaces

Direct sum of subspaces

Definition

Remarks

Reading path

Where this leads

Further reading

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