  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.
