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Linear Algebra for Data Science Using Python

Group and Field

Explore the fundamental concepts of groups and fields within vector spaces. Understand the axioms that define groups, including closure, associativity, identity, and inverse elements. Learn about abelian groups and the properties that characterize fields, such as additive and multiplicative groups and distributivity. This lesson helps you grasp these essential algebraic structures used in data science and prepares you to apply them in practical scenarios involving vectors and matrices.

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Group

A group G, over a binary operation, oo, and two elements a,b∈Ga, b \in G, is defined with the following four axioms:

  • Closure: A group is closed under oo, that is, the result of the operation is also a member of that group.

a o b=c∈Ga\ o\ b = c \in G

  • Associativity: The result of a binary operation on three or more elements remains the same, regardless of the arrangement of parentheses.

(a o b) o c=a o (b o c),∀ c∈G(a\ o\ b)\ o\ c = a\ o\ (b\ o\ c), \forall\ c \in G

  • Identity: There exists an identity element, ii, under operation, oo such that operation between any element of the group, cc, and ii results in c
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