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Definition of Vector space: Suppose that $F$ is a field, such as $\RR$ or $\CC$, for example. An $F$-vector space (for example, a real or complex vector space) is an ordered triple $(V, +, \cdot)$, where:
- $V$ is a set; the elements of $V$ are called vectors.
- $+$ is a binary operation on $V$, usually called vector addition.
- $\cdot$ is a function from $F \times V$ to $V$, usually called scaling or scalar multiplication.
These operations are required to satisfy the following axioms:
- The pair $(V, +)$ satisfies the axioms of an abelian group.
- Scalar multiplication is associative, in the sense that, for all $x,y \in F$, and all $v \in V$, $x \cdot (y \cdot v) = (x \cdot y) \cdot v$.
- Scalar multiplication is distributive, in the sense that, for all $x,y \in F$, and all $v \in V$, $(x + y) \cdot v = (x \cdot v) + (y \cdot v)$.
- Scalar multiplication is distributive, in the sense that, for all $x \in F$, and all $v,w \in V$, $x \cdot (v + w) = (x \cdot v) + (x \cdot w)$.
The following statements and definitions logically rely on the material of this page: Def/Basis (vector space), Def/Dimension, Def/Euclidean metric, Def/Linearly independent, Def/Span, Def/Subfield, Def/Subspace, and State/Intersections of subspaces are subspaces
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