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When I was a student (in the 1970s) I was taught linear algebra as an "adjunct" to "engineering mathematics" such as differential equations. That was during my sophomore year, which seems a bit late, but those were the days when "calculus" was all the rage.

Nowadays, the application emphasis is on information technology, which stores information in matrices, strings, and vectors. Given this emphasis should linear algebra be taught earlier, concurrently with computer science? Would that mean making it "simultaneous" with first year calculus or even pre-calculus?

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  • $\begingroup$ (This pertains to the U.S.) Precalculus sounds way too early to me, and the vast majority of precalculus students will never later take linear algebra anyway. However, I think above average calculus students could easily take linear algebra simultaneously with Calculus 2, and in fact the majority of the linear algebra students I taught in Fall 1999 and Spring 2000 were taking Calculus 2 (most with me, in fact) at the same time. This was an exceptional case though, as I think most colleges/universities (in the U.S.) require calculus 2 as a prerequisite for elementary linear algebra. $\endgroup$ – Dave L Renfro Jul 10 '14 at 19:34
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    $\begingroup$ @DaveLRenfro: "After Calculus 2" strikes me as the LATEST that people should study linear algebra. That is, it should precede multivariable calculus, where people have to know vector and matrix theory anyway, but where they get "linear algebra" piecemeal. $\endgroup$ – Tom Au Jul 10 '14 at 19:36
  • $\begingroup$ I was just have this discussion with another grad student today. I think it would act as a great introduction to mathematical thinking with basic (easy) proofs (bases, linear dependence/independence, rank-nullity, etc) that are applicable to most college freshmen that would perhaps increase understanding of proof techniques and mathematical thought in calculus (if taken simultaneously or after linear). I'm interested if there is any evidence of this effect of increased performance or if there has been serious consideration of this change. $\endgroup$ – Chris C Jul 10 '14 at 19:36
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    $\begingroup$ Additionally, I stumbled across this article that suggests linear algebra as an alternative to calculus for college freshmen: nctm.org/about/content.aspx?id=28195 $\endgroup$ – Chris C Jul 10 '14 at 19:39
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    $\begingroup$ In rereading your question it occurs to me that by linear algebra you might mostly mean work with matrices, and not the abstract idea of a linear function that can be represented (after ordered bases have been fixed for the domain and range) as a matrix and the corresponding mathematics behind this identification. If we're talking about mostly matrices and determinants and linear equations, then this can certainly be part of precalculus, and in fact college algebra and precalculus texts often devote one or two chapters to the topic (but it's usually not covered, in my experience). $\endgroup$ – Dave L Renfro Jul 10 '14 at 19:41
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My personal opinion. Anytime is a good time to teach linear algebra. I'd like to see at least two courses in the curriculum. Ok, more accurately:

  1. elementary matrix theory: solving systems, matrix math, determinants, routine computations of all sorts. Very limited proof emphasis. Here we are merely rounding out a computational core to give better tools to later calculus and other courses. This could be taught before calculus, in parallel with precalculus if need be. I would hope to include some appreciation of spanning and linear independence here. We'd like to use these terms intelligently in other courses before they hit linear algebra. Applications abound in this course. For example, Lay's Linear Algebra would be ideal.
  2. linear algebra: focus on proofs and abstraction. Includes abstract vector spaces, subspace and basis theorems. Either this course comes after the introduction to proofs course and gets to the Jordan form, or, the topics covered are narrowed considerably and the course is used as the introduction to formal proofs course. For the non-introductory to proofs version, you might use Insel, Spence and Friedberg

There are good reasons for making this demarcation in courses. The elementary matrix theory course is really worthwhile for almost every technical major and if it was earlier in the degree plans then many sections could be offered at a larger school making scheduling easy for the student. On the other hand, linear algebra proper is really just a course for math majors and it is an absolute drudgery to try to teach proofs to engineers. I speak with considerable experience.

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The way "programs store data in arrays" has next to nothing in common with linear algebra. Don't mix them up, the superficial similarities are confusing enough as is.

For a nice look at what linear algebra is really about check out Treil's "Linear Algebra done Wrong".

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