Alternate Definitions of Vector Space Dimension and Module Rank Using Isomorphisms

TL;DR

This paper introduces new definitions for vector space dimension and module rank based on isomorphisms, simplifying the concepts by avoiding bases and linear independence, and offering an alternative educational perspective.

Contribution

It proposes isomorphic definitions for dimension and rank that do not rely on bases, enabling more straightforward proofs and teaching methods in linear algebra and module theory.

Findings

Isomorphic dimension equals n if a linear isomorphism exists from F^n to V.

Isomorphic rank equals n if an R-module isomorphism exists from R^n to M.

The approach simplifies proofs and enhances educational methods in algebra.

Abstract

The standard definition of the dimension of a vector space or rank of a module states that dimension or rank is equal to the cardinality of any basis, which requires an understanding of the concepts of basis, generating set, and linear independence. We pose new definitions for the dimension of a vector space, called the isomorphic dimension, and for the rank of a module, called the isomorphic rank, using isomorphisms. In the finite case, for a vector space $V$ over field $F$, its isomorphic dimension is equal $n$ if and only if there exists a linear isomorphism from $F^n$ to $V$. For a module $M$ over the commutative ring $R$ with identity, its isomorphic rank is equal to $n$ if and only if there exists an $R$-module isomorphism from $R^n$ to $M$. There are similar definitions in the infinite cases. These isomorphic definitions do not require the concepts of basis, generating set, and linear independence. This approach allows for some fundamental linear algebra and module theory results to be seen more easily or to be proven more similarly to other algebraic proofs involving isomorphisms and homomorphisms and provides an alternate educational approach to dimension and rank.