vector space

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

  1. $ (\u +\v )+\mathbf{w}= \u +(\v +\mathbf{w})$ for all $ \u ,\v ,\mathbf{w}\in V$
  2. $ \u +\v =\v +\u$ for all $ \u ,\v\in V$
  3. There exists an element $ \mathbf{0} \in V$ such that $ \u +\mathbf{0}=\u$ for all $ \u\in V$
  4. For any $ \u\in V$ , there exists an element $ \v\in V$ such that $ \u +\v =\mathbf{0}$
  5. $ a \cdot (b \cdot \u ) = (a \cdot b) \cdot \u$ for all $ a,b \in F$ and $ \u\in V$
  6. $ 1 \cdot \u = \u$ for all $ \u\in V$
  7. $ a \cdot (\u +\v ) = (a \cdot \u ) + (a \cdot \v )$ for all $ a \in F$ and $ \u ,\v\in V$
  8. $ (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 $ \mathbf{0} \in V$ is called the zero vector of $ V$ .

This entry is a copy of the GNU FDL vector space article from PlanetMath. Author of the original article: djao. History page of the original is here



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As of this snapshot date, this entry was owned by bloftin.