# Why is it called "linear" algebra?

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I tell non-math I'm studying linear algebra and they say: "that sounds like something I took in sixth grade."

So why call it linear algebra? It is supposed to be in contrast to "abstract algebra" ?

Thanks

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Linear algebra develops from techniques used to solve systems of linear equations. Building on these methods, linear algebra gets into the study of vector spaces, which are sets of vectors combined with two operations such that certain requirements (which we call axioms) are met.

Abstract algebra is the study of algebraic structures (including vector spaces, but also groups, rings and others) and thus serves as a generalization of all this, i.e. it deals with the properties of various sets of elements combined with various operations following various axioms.

Edited by John
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I tell non-math I'm studying linear algebra and they say: "that sounds like something I took in sixth grade."

Then they are greatly underestimating it. If they tease you, take a question from an old exam and tell them to work the problem. That'll shut 'em up

So why call it linear algebra? It is supposed to be in contrast to "abstract algebra" ?

I have yet to reach abstract algebra, but it seems to have a totally different feel to it than linear algebra, just by looking at some open-source material. I do know they are completely separate courses in university, and math majors will take abstract some time after linear.

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As John says, the theory developed from the study of linear equations and vector spaces. To add a geometric prespective here, linear algebra is really all about the study of matrices, which one should think of as linear transformations between vector spaces. So, for example, a linear transformation on $\mathbb{R}^{n}$ is of the form

$x^{a'} = x^{b}T_{b}^{\;\; a'}$,

where $T_{b}^{\;\; a'}$ is a real matrix of dimension nxn. Explicity you see that the transformation is "just" a linear change of coordinates, x appears on the right only once. (I have used some conventions here about rows and columns, but you get the picture)

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At least you aren't studying number theory - that's something they did in second grade.

My own response, back in the day, was to agree that I should have paid better attention in sixth grade or whenever. That had the advantage of being both true and not high-handed.

But another response (and this works for a variety of subjects and social contexts, if your field of study is an honest one and you are dealing with social gamesmanship of the simple kind) might be to say what you are actually, specifically working on at that moment - which would not be the general label of the large topic, but something like "the eigenvalues of finite dimensional rotation matrices".

And there's a side effect: being able to say clearly what you are doing at the present time can be informative, even enlightening, for oneself.

Edited by overtone
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• 2 weeks later...

So why call it linear algebra? It is supposed to be in contrast to "abstract algebra" ?

Just to answer your last statement. Abstract Algebra deals with the study of mathematical structures called groups. To give an example between LA (Linear Algebra) and AL (Abstract Algebra) on their similarity (not in property, but on the approach, since it's algebra).

In LA we have basis sets that spans a particular vector space, and how an entire vector space can be constructed by the basis set. Similarly in AL we have cyclic sets that generates an entire group! So their approach is similar, but as John puts it:

Linear algebra develops from techniques used to solve systems of linear equations. Building on these methods, linear algebra gets into the study of vector spaces, which are sets of vectors combined with two operations such that certain requirements (which we call axioms) are met.

that's why is called Linear Algebra.

Besides, LA has immense applications; handwriting analysis, solving linear first-order differential equations, search engines use the mathematics from LA, Differential Geometry: representation of the coefficients of the first-fundamental form is in matrix form.

Hope this clarifies things.

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Just to answer your last statement. Abstract Algebra deals with the study of mathematical structures called groups.

Abstract algebra means more than just groups, but yes groups are an important part of abstract algebra. But so are rings, fields, monoids, semigroups, modules, vector spaces, algebras and Lie algebras. (And things I am sure I have missed).

One can think of these as sets with extra structures on them.

All these peices from the mathematical backbone of just about everything.

Edited by ajb
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• 2 months later...

Why are they Laughing at it, it is very important, I guess they do not know what are its uses. It is very useful to Engineers even though it is basic, it simplifies everything. Even Newton I think used it before to derive equations for calculus as well as leibnitz.

Edited by gabdecena
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Even Newton I think used it before to derive equations for calculus as well as leibnitz.

That would have quite the feat of precognition on Newton's part. Linear algebra didn't exist in Newton's time. While Leibniz (but not Newton) did use determinants, it wasn't really linear algebra that he was using. Linear algebra got it's start with Vandermonde in the late 18th century. The vectors we use now in physics weren't invented until the late 19th century. The rather abstract extensions such as Hilbert spaces -- those are 20th century developments.

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