Vector Space

Learn the definition of vector space and its axioms and see examples.

Definition

A vector space, VV, defined over a field, FF, is a set with addition (++) and scalar multiplication (×\times) operations, obeying the following axioms.

Note: For the axioms below, consider x,y,z,0,xˉV\bold{x,y,z,0,\bar{x}}\in V and α,β,1F\alpha,\beta,1 \in F.

  • Closure:
  1. (x,y),      x+yV\forall\bold{(x,y)},\;\;\; \bold{x}+\bold{y}\in V
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  • Additive identity
  1. 0 x,      x+0=x\exist\bold{0 }\ \forall\bold{x},\;\;\; \bold{x+0=x}
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