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Linear Algebra for Data Science Using Python

Vector Space

Explore the core concepts of vector spaces by learning their defining axioms, including closure, associativity, and distributivity. Understand various examples such as real coordinate spaces, matrices, functions, and polynomials, and see their significance in data science.

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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,xˉ∈V\bold{x,y,z,0,\bar{x}}\in V and α,β,1∈F\alpha,\beta,1 \in F.

Vector space axioms

  • Closure:
  1. ∀(x,y),      x+y∈V\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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