Basis (linear algebra)/Advanced

A basis for a unital module $$M$$ over a ring with identity is a subset $$A$$ of $$M$$ such that every element of $$M$$ can be written uniquely as a finite linear combination of elements of $$A$$. The module $$M$$ is said to be free on the set $$A$$. When $$R$$ is a division ring, $$M$$ is called a vector space.

A basis for a module can be considered as the building blocks from which all elements of the module can be constructed. This is analogous to viewing as the prime numbers as building blocks from which all whole numbers can be assembled. Let us make this more precise. The abelian group $$\mathbb{Q}^{+}$$ of positive rational numbers under multiplication is a module over the integers. The prime numbers are a basis for $$\mathbb{Q}^{+}$$: every positive rational number can be written uniquely as a finite product of integral powers of prime numbers (and such factorization or a rational number corresponds to the unique lowest terms representation of the number as a fraction).

While every nontrivial vector space has a basis, not every module over an arbitrary ring will have a basis. Those that do have a basis are called free modules. There are other important characterizations of free modules, which can be found at the page on free modules.

Alternative definition of basis
Above, a basis for a module was defined as a subset of the module such that every element of the module can be uniquely written as a finite linear combination of elements of the module. An equivalent definition of a basis for a module is a subset of the module that spans the module and is linear independent.

The invariant dimension property
Every basis for a vector space has the same cardinality. This is not true for free modules over arbitrary rings. A ring for which any two bases for a free module over the ring have the same cardinality is said to have the invariant dimension property.