Linear and multilinear algebra; matrix theory Definition

Sum of subspaces

Definition

Let $F$ be a field, let $V$ be a vector space over $F$ , and let $U$ and $W$ be linear subspaces of $V$ . The sum of $U$ and $W$ is $U + W = \{u + w : u \in U, \ w \in W\}.$ For subspaces $U_1, \dots, U_k$ , the sum is $U_1 + \cdots + U_k = \{u_1 + \cdots + u_k : u_i \in U_i\}$ .

Remarks

The sum $U + W$ is a subspace, and it is the smallest subspace of $V$ that contains $U \cup W$ . The sum is the span of $U \cup W$ . When $U \cap W = \{0\}$ , the sum is a direct sum. For finite-dimensional subspaces, Grassmann's formula gives $\dim(U + W) = \dim U + \dim W - \dim(U \cap W)$ .

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Vector space definition
Linear subspace definition
Sum of subspaces definition

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Tag	`0RMKABS2S5`
URI	`m2://0RMKABS2S5`
Kind	definition
Writing	W1 stated
MSC	15A03
Defines	Sum of subspaces
Uses	Linear subspace

Sum of subspaces

Definition

Remarks

Reading path

Where this leads

Further reading

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