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vector space
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(Definition)
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Let $F$ be a field (or, more generally, a division ring). A vector space $V$ over $F$ is a set with two operations, $+: V \times V \longrightarrow V$ and $\cdot: F \times V \longrightarrow V$ , such that
- $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in V$
- $\u+\v=\v+\u$ for all $\u,\v\in V$
- There exists an element $\0 \in V$ such that $\u+\0=\u$ for all $\u \in V$
- For any $\u \in V$ , there exists an element $\v \in V$ such that $\u+\v=\0$
- $a \cdot (b \cdot \u) = (a \cdot b) \cdot \u$ for all $a,b \in F$ and $\u \in V$
- $1 \cdot \u = \u$ for all $\u \in V$
- $a \cdot (\u+\v) = (a \cdot \u) + (a \cdot \v)$ for all $a \in F$ and $\u,\v \in V$
- $(a+b) \cdot \u = (a \cdot \u) + (b \cdot \u)$ for all $a,b \in F$ and $\u \in V$
Equivalently, a vector space is a module $V$ over a ring $F$ which is a field (or, more generally, a division ring).
The elements of $V$ are called vectors, and the element $\0 \in V$ is called the zero vector of $V$ .
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"vector space" is owned by djao.
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Cross-references: vectors, ring, module, element, operations, division ring, field
There are 483 references to this entry.
This is version 12 of vector space, born on 2001-10-19, modified 2006-07-22.
Object id is 364, canonical name is VectorSpace.
Accessed 66806 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) | | | 20-00 (Group theory and generalizations :: General reference works ) | | | 13-00 (Commutative rings and algebras :: General reference works ) | | | 16-00 (Associative rings and algebras :: General reference works ) |
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Pending Errata and Addenda
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