# What is the Definition of Linear Algebra?

For introducing a field of science, usually we require a definition which summarizes the goal which is intended to be achieved by this field. Linear Algebra is one of the most important parts of mathematics which has not only interesting pure mathematical ideas but also a lot of applications in physics and engineering. So it seems reasonable to have a Complete, Clear, Brief, and Delicate definition for this important branch of mathematics.

I am wondering that what is the best fit for such a definition. For example, the very first sentence that the Linear Algebra Done Right by Sheldon Axler starts with is

Linear Algebra is the study of linear maps on finite dimensional vector spaces.

What is your definition of Linear Algebra in few sentences?

• Hard to improve on Sheldon Axler's – AdLibitum Aug 1 '16 at 11:45
• @AdLibitum: Hah! :D Yes, you are right! But we may wait to see what others see in linear algebra! :) – Hosein Rahnama Aug 1 '16 at 11:46
• I reject the limitation to "finite dimensional". Even in undergraduate mathematics, we see an interaction between linear algebra and differential equations. – GEdgar Aug 1 '16 at 13:51
• I don't think such a definition is useful in any field of science. – egreg Aug 1 '16 at 15:31
• The study of solving systems of linear equations? – Samasambo Aug 18 '16 at 21:42

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.

• “Physics is what physicists do late at night” (Jay Orear). – egreg Aug 1 '16 at 15:29
• Would it be accurate, in your opinion, to rephrase that as [X Theory] is the study of the category of [sets with the structure X]? – MphLee Dec 11 '19 at 16:57

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.

• Great definition indeed! I wanted to learn why we didn't learn much about infinite dimensional spaces when first starting out lower div lin alg, but after reading about the Hilbert spaces of functions, and doing corresponding analysis with signals, it seems much more as the basis of real analysis. However, I posit that some foundations on the theory of vector spaces, inner product spaces, etc, are still very much in play even if it is "analysis" and not linear algebra – Charlie Tian Jun 20 '17 at 15:34
• This definition together with this explanation is useful, at least as a pedagogical introduction to first year algebra students. – Miguel Feb 20 at 9:42