What is the Definition of Linear Algebra?

Some of the comments above wonder about my description of linear algebra as the study of linear maps on finite-dimensional vector spaces. “Finite-dimensional” is specified because the deep and exciting properties of linear maps on infinite-dimensional vector spaces require that analysis be brought into the picture. This moves the subject from linear algebra to functional analysis.

For example, in infinite-dimensions deeper results are available on Banach spaces than on more general normed vector spaces for which Cauchy sequences might not converge. As another example, orthonormal bases in Hilbert spaces are used in connection with infinite sums.

The deep properties of linear operators on finite-dimensional vector spaces, such as the existence of eigenvalues, the singular-value decomposition, and so on, either do not have good analogs on infinite-dimensional vector spaces or use much different techniques (and lots of analysis). Thus it makes sense to think of linear algebra as the study of linear maps on finite-dimensional vector spaces. When analysis needs to be added to the theory, “functional analysis” seems to be a better term to describe the subject.

Essentially all of the major parts of abstract algebra have the same definition.

[X Theory] is the study of [sets with the structure X] and the [whatever homomorphisms are called in this subject] between them.

For example:

Linear Algebra is the study of vector spaces and the linear maps between them.

Group Theory is the study of groups and the group homomorphisms between them.

Lattice Theory is the study of lattices and the lattice homomorphisms between them.

Note that as usual in math, the definitions themselves don't tell you very much. You have to actually study these topics to understand, for instance what a vector space is, what a linear map is, and why these ideas are useful.