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January 24[edit]

Why does lineal algebra have such a central role in many, if not all, math degrees?[edit]

Math degree programs seems to include invariably Calculus (single and multivariate), probability, and lineal algebra. Then, you can find more applied fields like game theory, analysis of algorithms. Why not universal algebra, or, abstract algebra? Wouldn't these be more basic than lineal algebra? --Senteni (talk) 14:46, 24 January 2015 (UTC)[reply]

From the US university programs I'm familiar with, the Calculus (and differential equations), probability, and linear algebra classes you mentioned are expected to be completed at the underclassman (freshman/sophomore) level, and may be all the math that is offered at a two-year junior college. Math majors will be expected to take a full year of abstract algebra (as well as topology and real analysis) as upperclassmen (juniors/seniors), as well as additional in-major electives. So I don't see linear algebra playing the the central role you describe, although it provides an introduction to some of the elements of abstract algebra, and is often the first class in which students are expected to be able to write formal proofs (aside from the delta epsilon proofs that may still be required in some calculus classes). This is the first roadblock in the studies of some math majors, and I know a few who sailed through calculus, but changed majors after having trouble in linear algebra and realizing that it was unlikely that they would make it through an abstract algebra class. -- ToE 18:07, 24 January 2015 (UTC)[reply]

(ec) "We share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury." (Irving Kaplansky, writing about himself and Paul Halmos).

Why study linear algebra? Because it gets everywhere. On the applications side: quantum mechanics -- linear algebra; co-ordinate transformation locally -- linear algebra (globally too, if it's flat); basis function decompositions -- Fourier theory, Laplace theory, wavelets, functional analysis, orthogonal function theory -- all linear algebra. Big multidimensional simulations -- eg numerical weather forecasting, econometrics -- mostly (big) linear algebra. Engineering applications -- stability theory, signal processing -- mostly linear algebra. Linear calculations can be big and fast; even for nonlinear problems, the solution of choice will often go through solving a sequence of linearised steps. And the linearisation may be very revealing about the nature of the solution.

On the purer side, too: every group has a linear representation (group representation theory), leading to a lot of pure maths (as well as a fair number of applications), exploring group theory (and deeper extensions of it) in a systematic way. So there is a lot that you can represent with matrices. And even if you're interested in more abstract properties of algebra, it's useful to be able to illustrate them in a concrete way, or show phenomena in simple toy systems. Matrices can often provide that -- whether it's simply introducing the idea of non-commutativity, all the way to really quite hairy stuff.

So yeah, there's a reason that linear algebra is right up there with calculus and probability. Jheald (talk) 18:30, 24 January 2015 (UTC)[reply]

I clearly poorly expressed myself. I mean "central" in the sense that it's basic, not that it is the learning object. Do many things build upon it? Couldn't the programs have been organized like the Algebra book of Serge Lang,where Lineal Algebra just shows up on part III? His book puts Groups, Rings, Modules, Galois theory, Fields before the Matrix and the like. --Senteni (talk) 23:54, 24 January 2015 (UTC)[reply]

Well, not really. More general does not necessarily equate to "better". For example, the entire field of representation theory exists because linear algebra is "easy", whereas group theory is "hard". Basically all non-trivial problems in group theory involve realizing the groups involved as matrices in some fashion. But that's just in abstract algebra. You can work in other fields like analysis or dynamics, and never encounter a "ring" or "group", yet use linear algebra regularly. There one studies linearizations of problems, again because linear things are easy to do, but non-linear things are hard. In a sense, this is also what differential calculus is about too. So, yes, you need linear algebra to do just about anything in mathematics. Whereas group theory and ring theory, not so much. Sławomir Biały (talk) 00:41, 25 January 2015 (UTC)[reply]

It is linear with an r at the end. Yes lots of things build on it, in fact it has more direct applications around mathematics than anything else, probably the only places where you don't use it would be in areas like number theory or set theory. See special linear group for instance for just one application in group theory. Dmcq (talk) 01:06, 25 January 2015 (UTC)[reply]

In Spanish it is álgebra lineal. -- ToE 21:12, 25 January 2015 (UTC)[reply]

Senteni, I think you expressed yourself well enough that I was the only one who misunderstood you question. The others have pointed out the usefulness of linear algebra on its own, but I add that the concepts taught there ease the way into algebra for a lot of students. There are those who would do well jumping immediately into the abstract presentations of the year long undergraduate abstract algebra class, but a lot would have problems. Note Cal's Prerequisites to Math113 Introduction to Abstract Algebra is "Math 54 (Linear Algebra and Differential Equations) or a course with equivalent linear algebra content". Tangentially related to this is the use of concrete examples and exercises in an abstract algebra course. I've seen Dummit and Foote's Abstract Algebra used as an undergraduate text in universities with a strenuous program and as a graduate text in those with a less strenuous one. I would highly recommend it to anyone taking a class based on a more abstract text, such as Lang's Algebra, purely for its large number of examples and exercises.

Back to linear algebra, the one semester sophomore level class is only a brief introduction to the field, and a more in-depth Junior/Senior level course should be available as an in-major elective to most math majors (such as Cal's Math 110 Linear Algebra, and that linear algebra will be a large component of a lot of other courses. -- ToE 19:52, 28 January 2015 (UTC)[reply]