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All of the concepts and terminology central to linear algebra were established in the late 19th century.

Following recent comments by user KCd, that in the early 20th century determinants were the “primary language of linear algebra”, I conclude (rightly or wrongly) that linear algebra originally focused on techniques for solving systems of linear equations and the role of matrices and their determinants.

Today, the focus of linear algebra is the study of vector spaces and their properties.

Who were the mathematicians that brought about this change of emphasis? Was there a particular paper or textbook, or was this change simply the result of a natural move towards a more abstract presentation of the subject?

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Like mathematics in general, Linear Algebra is evolving. Vector spaces were introduced by Sylvester in the 19th century and became a core of Linear Algebra University courses by Halmos ("Finite dimensional vector spaces", 1947) and Gelfand ("Lectures in Linear Algebra", 1948). Motivation came from physics (quantum mechanics in particular).

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  • $\begingroup$ I have been looking at it from the point of view of an undergraduate level introduction to the subject. The subject certainly continues to evolve. As far as I understand, the subject has seen considerable renewed interest in recent years. I have always believed the Grassmann introduced vector spaces - why Sylvester? The texts which you have references certainly appear to get stellar reviews online. The Halmos text is described as giving the first modern axiomatic presentation, so these look like good candidates for notable early modern textbook presentations. $\endgroup$
    – nwr
    Mar 27 at 21:09
  • $\begingroup$ As often happens it is impossible to name exactly who introduced a math concept. Some sources name Sylvester, some name Grassmann. Halmos and Gelfand were excellent teachers. Gelfand recommended his book to undergraduates and even to strong high school students (I witnessed that myself). He compared his book with Lang's algebra by saying that "gasoline has more calories than butter but you eat butter, not gasoline". $\endgroup$
    – markvs
    Mar 27 at 21:34
  • $\begingroup$ That's a great quote. I wonder if Gelfand ever tried a vindaloo - it is full of butter and has the same explosive effect as gasoline. $\endgroup$
    – nwr
    Mar 27 at 22:27
  • $\begingroup$ A number of online sources cite these two texts as key to the formulation of modern linear algebra - especially that of Halmos. $\endgroup$
    – nwr
    Apr 2 at 23:47
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Actually, I think it is not quite the case that what we now think of as "linear algebra" (either terminology or canon of standard results) existed in the late-ish 19th century.

Yes, "matrices" were about determinants... Yes, people knew how to solve (finite) linear equations.

It seems that the most serious impetus to "linear algebra" (linear operators, eigenvalues...) was from infinite-dimensional situations, arising in the integral equations' equivalent versions to solving differential equations, as in Hilbert, Schmidt, et al, c. 1895. Yes, Sturm-Liouville results and ideas (c. 1835) were pose-able in linear algebra, but the terminology did not exist then. In fact, their wonderful ideas were not made rigorous (or even made precise) until Steklov and Bocher after 1890.

Yes, the idea many people (including myself) had/have of the history of linear algebra is that finite-dimensional stuff came first, and then fancier Hilbert-space stuff, ... but apparently that is not at all true! Crazy! :)

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    $\begingroup$ Thanks for your answer. I didn't mean to suggest that linear algebra was a well-formed subject by the late 19th century. Only that Grassmann had introduced the concept of a vector space and that key concepts such as linearity, linear independence, linear combination, bases, span, matrices, determinants, etc... were all established then. The extension to infinite-dimensional spaces which you highlight is clearly key to the subject achieving maturity. As far as I understand, the subject did not really take its modern form until sometime after the early/mid 20th century. $\endgroup$
    – nwr
    Mar 26 at 17:10
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This may count as a partial answer, I don't claim to have the whole story.

The metaphor of vector spaces is currently being included especially among teaching tools to encourage learners and users of linear algebra and matrix problems to think of and visualise (or almost-visualise) the involved matrix and vector objects in an enlarged variety of ways.

A notable and much-appreciated exponent of this teaching approach is apparent in Prof. Gilbert Strang of MIT. His approach appears especially through his textbooks and lectures especially his introductions to linear algebra.

I've no information about whether Prof. Strang had any close predecessors in this approach or not, but he certainly appears to engage in numerous innovative approaches to the teaching of his subject, and might indeed be at the origin of the tendency noted in the question.

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  • $\begingroup$ Strang is an interesting suggestion. Coincidentally, I recently watched his 2020 opencourseware series on linear algebra and he certainly presented an original method for teaching the subject. (Also a rather charming style if I recall correctly.) Still, I can't help but feel that he had predecessors when it comes to the emphasis of vector spaces. $\endgroup$
    – nwr
    Mar 26 at 3:53
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    $\begingroup$ Strang's first book was published in 1976, and by this time (in the U.S.) even lower level undergraduate courses in linear algebra (LA) were common (I read through about half of the 1970 1st edition of Bernard Kolman's Elementary Linear Algebra in 1975, where it was used for a heavily populated course at a nearby university; the study of abstract vector spaces begins with Chapter 2 on p. 52). I think upper level undergraduate LA courses were phased in during the 1950s and early 1960s at most of the larger U.S. universities. (continued) $\endgroup$ Mar 26 at 7:28
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    $\begingroup$ Note that the well known book by Halmos was published by D. Van Nostrand in 1958 (the earlier 1942 version was probably not very widely used, as it was Princeton University's Annals of Mathematics Studies #7) and the still widely used book by Hoffman/Kunze appeared in 1961. I have several similar books on my bookshelves, such as by Stoll/Wong (1968) and Ames (1970). I think the evolution of the lower level introductory LA courses began in the late 1960s, and became widespread in the 1970s (continued) $\endgroup$ Mar 26 at 7:30
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    $\begingroup$ when math departments were finding at this time the need (due to a greater diversity of student backgrounds) for transitional courses from computational calculus work to proof-based upper level work. Thus, by the time Strang's first book appeared, both levels of LA were (in the U.S., at least) were very widely taught. See this answer for more specifics about "my classifications" of U.S. LA courses. $\endgroup$ Mar 26 at 7:30

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