Vector Spaces in Linear Algebra vs Abstract Algebra

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In Linear Algebra, the scalars for a vector space are typically either real or complex numbers. In Abstract Algebra, the scalars for a vector space can come from any field, including a finite field. In Abstract Algebra, a vector space V over a field F is, first, an Abelian group under vector addition (the elements of V are called vectors and the + sign refers to adding vectors). Secondly, there is a scalar multiplication of any element of F times any element of V, giving us an element of V. More precisely, the scalar multiplication is a mapping F × V → V (F × V is the Cartesian product of F and V). It also satisfies associative and distributive properties, as well as an identity property with respect to the unity 1 ∈ F. In this abstract setting, we can still describe spanning sets, linear independence, basis, and dimension, based on the invariance of the size of a basis. But some vector spaces are infinite-dimensional.

#LinearAlgebra #AbstractAlgebra #VectorSpace #VectorSpaces

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