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# Def/Vector space

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:

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 $v,w \in V$, $x \cdot (v + w) = (x \cdot v) + (x \cdot w)$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Ordered tuple, Def/Binary operation, Def/Abelian, Def/Group

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

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Linear algebra, Clust/Abstract algebraic structures